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Cunha scored twice as the Red Devils came back from two-goals down in the Champions League to potentially save Carrick's job.
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However, the Brazil forward was sent off in the dying moments as United clung onto their first win in ten matches.
Having already been booked for taking his shirt off, Cunha was shown a second yellow for a late challenge moments after the 86th-minute winner.
The 27-year-old did not get very far down the tunnel and quickly re-emerged to join in the post-match celebrations after the full-time whistle.
But TV cameras picked up an interaction with Carrick which appeared to stop Cunha in his tracks.
The Brazilian was seen remonstrating furiously as a club official took him back down the tunnel.
At one point, Cunha shouted angrily as he reluctantly made his way back to the dressing room without his teammates.
He then stopped to applaud the fans above the tunnel before leaving the pitch.
According to IFAB's Laws of the Game, players must leave 'the vicinity of the field of play and the technical area' once dismissed.
Meanwhile, the referee still has authority over the match event in the immediate moments after the final-whistle.
While exceptions are often made for cup finals and trophy-lifts, Cunha was technically not allowed anywhere near the pitch while the officials were still there.
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It is not clear whether this was the reason behind Carrick's decision to order him back down the tunnel.
A desperate United side found themselves 2-0 down inside 17 minutes through goals from Matias Soule and Donyell Malen.
Carrick's men failed to have a shot on target during the first-half and left the field to a chorus of boos from a furious Old Trafford.
However, they came roaring back in the second-half as Bruno Fernandes scored a retaken penalty before Cunha's late double.
While there was an element of fortune about his first goal following a failed Roma clearance, his second was wonderfully converted after a defence-splitting pass from teenager Shea Lacey.
It sparked jubilant scenes as a shirtless Cunha jumped into the crowd, although that would ultimately contribute to United being reduced to ten men.
A reckless challenge followed five minutes later - and Carrick would be forgiven for being a little aggrieved with his match-winner over a completely avoidable red card as United battled for a precious three points.
The win relieves the pressure on Carrick, for now, with United set to host Aston Villa in the Premier League on Sunday before the international break.