Mourinho's protests proved too much for the officials as the Portuguese was ushered to the stands at Old Trafford
United will fervently hope the decline is only temporary as they now stand eight points adrift of co-leaders Manchester City, Liverpool and Arsenal.
Ibrahimovic's best moment was a trademark overhead kick that was brilliantly saved by Heaton, its power injuring the goalkeeper.
Other chances inside the box came to nothing, including a that drifted on to the bar. In the final minute, Paul Pogba squeezed a cross to the super Swede. The 13-time league champion fluffed his lines, missed the target and held his head in shame.
For someone of his talent and ego, it was a sad moment.
United striker Zlatan Ibrahimovic shows his frustration after another good chance goes begging against the Clarets
The former Paris Saint-Germain striker is thwarted by the feet of keeper Tom Heaton on his return to his former club
Heaton was in inspired form and determined to keep hold of his clean sheet with a string of impressive saves
United captain Rooney returned to the substitutes' bench for his side following injury and came on after the break
United will wonder how they allowed Burnley to leave Old Trafford with their first point of the season even though they played the final quarter with only 10 men following the unfortunate dismissal of Ander Herrera.
Certainly, part of it was down to the brilliant former United trainee Heaton who saved from stand-in captain Juan Mata and Ibrahimovic within the first five minutes, and followed up with worldies from Jesse Lingard and Ibrahimovic again.
But there was also a lack of composure and confidence in United's ranks when the chances came. Burnley's Mancunian defenders Michael Keane and Ben Mee were heroic in the heart of their defence but they aren't Carlos Puyol and Fabio Cannavaro, guys who Zlatan has played with and against.
Captain for the day Juan Mata protests to referee Mark Clattenburg at half-time as the home side were held at the interval
£89million signing Paul Pogba looks to turn the screw on Sean Dyche's side but the visitors stood firm throughout the match
Burnley striker Andre Gray returned to the side following his four-match suspension and he provided Daley Blind with issues
Ander Herrera goes into the book for a late and rash challenge on Burnley midfielder Dean Marney during the first-half
Lingard struck the post and United had three loud penalty appeals waved away, the first for a tug by Jon Flanagan on Matteo Darmian that incensed Mourinho so much he vented at referee Mark Clattenburg in the tunnel at half-time and was sent to the stands for the second 45 minutes.
Rui Faria tried to direct operations in his boss' absence – though assistants were seen running to and from the directors' box with pieces of paper – and afterwards tried to put a brave face on a result that would have seen Louis van Gaal get slaughtered.
'We always believed the goal was going to come. When the best player is their goalkeeper, you feel you will get one goal minimum,' he said.
'The boys will have better days for sure. Happy days. They must keep believing. We kept fighting with 10 men as much as 11.'
Ibrahimovic finds himself hounded out by two Burnley defenders as he looked to break his run without a goal for United
Jesse Lingard was selected as part of United's creative engine but he struggled to break down a resolute Burnley defence
Mourinho watches on glum-faced from the directors' box during the second-half as United streamed forward for a goal
Mourinho is already in hot water with the FA following his comments made regarding Anthony Taylor earlier this month
The Burnley goal lived a charmed life at times and Mata was very close to putting United in front when he struck a post
United's unfamiliar line-up saw four left backs line up in defence – Darmian, Daley Blind, Marcos Rojo and Luke Shaw.
Most of the time the action was up the other end though, as Burnley survived with minimal possession though Icelandic international Johann Gudmundsson's shot at David de Gea with their best opening.
'Heaton was superb,' confessed their manager Sean Dyche afterwards. Of the Ibrahimovic effort where the goalkeeper made himself large and took the force of the shot before it deflected on to his leg and clear, Dyche joked: 'He's just spread himself, hasn't he.'
'I thought I'd broken my arm,' admitted the goalkeeper.
Herrera is already protesting his innocence as he falls to the ground falling his collision with the marauding Marney
Clattenburg, however, felt he had no choice but to brandish a second yellow card and red to the fallen, diminutive Spaniard
Dyche bellows out instructions to his players who had their backs against the wall for much of the game in Manchester
Heaton holds his hand after showing his bravery to deny United and preserve his clean sheet on his return to Old Trafford
United plodded through much of the opening 45 minutes. The one decent piece of interplay between Ibrahimovic and Pogba saw the Swede's shot blocked by the head of Mee.
Zlatan missed a one-on-one and Lingard's flick header was bound for the top corner until Heaton dived and clawed it away.
Herrera left his studs on Dean Marney's thigh and got a yellow – which was to be significant later on – and after 44 minutes, Mourinho went potty when Darmian ran across Flanagan into the box and went down as the defender put a foot and an arm across the Italian's wrist.
Mr Clattenburg, probably correctly, failed to point to the spot as the contact was minimal and initiated outside the box. Mourinho continued his argument down the tunnel at half-time and was officially dismissed.
He tried to take a seat in the front row of the stands in order to be close to the dug-out but was ordered higher up where he sat on the end of a row in the directors' box.
Faria admitted there was communication between them. 'Yes. Normally we define situations according the difficulties of the game. Sometimes a detail is important,' said the United No 2.
'But I don't think (the sending-off) was important to the result of the game.'
Indeed the second half was breathless even without Mourinho being a visible presence pitchside.
Mata's shot from Ibrahimovic's cutback was brilliantly stopped by Heaton from eight yards. Ibrahimovic's shot was blocked by Mee and Mata was then thwarted by Keane, Mr Clattenburg ruling the defender couldn't have got out the way even if it had hit his shoulder.
Manchester United coach Rui Faria took over on the sideline as Mourinho watched on powerless from the stands
Juan Mata and Jesse Lingard were replaced by Rooney and Marouane Fellaini but it failed to improve United's fortunes
Wayne Rooney came on as a substitute in a desperate bid to inject some creativity into the side despite 36 shots on goal
Rooney and Luke Shaw hold their hands to their face after a final chance comes and goes in second-half stoppage time
Darmian had a third penalty shout denied following a meaty challenge by Ashley Barnes and Ibrahimovic's acrobatic take-off didn't work from the full back's cross.
Even Herrera's red card, given when he slipped into Marney and got a second yellow, didn't stop United surging forward though Faria – standing in for Mourinho post-match – could have been impersonating his boss when he sarcastically remarked: 'Fantastic work from the referee.'
Rooney and Marouane Fellaini came on but Burnley's blue wall stayed firm, just.
'You could see there wasn't a lot left in the tank at the end,' said Heaton. Rooney skied the final opportunity from 20 yards and Old Trafford booed Clattenburg's signal that fie minutes of injury time were over.
Mourinho's actions will again prove a distraction from the result, United's third Premier League game without a goal.
But he will have plenty to think about in his lonely hotel room over the next few days. More than match officials, he needs his star No 9 to turn in a decent show.
