It forced his opposite number Steve Agnew - who was taking charge of Boro for the first time - to seek Mourinho out for the traditional post-match handshake.
Harry Redknapp wants the Middlesbrough job, according to a report on Saturday.
Wheras Karanka had, at times, criticised the club's supporters and the mindsets of certain players, Agnew, pointedly, praised both the fans - "they've been absolutely first class all season" - and the "attitudes and application" of "the best set of players I've worked with".
Gibson placed assistant head coach Agnew in charge of team affairs on Thursday after Spaniard Karanka's reign had come to a close. Stewart is somebody I've known a long time, so I know how he works. They never recovered from the episode last season that led to Karanka being absent from the touchline for Boro's 2-0 defeat at Charlton.
Jose Mourinho celebrated Antonio Valencia's late goal in Manchester United's 3-1 win over Middlesbrough by immediately heading to the dressing room pumping his fists.
'I couldn't be more proud of the group of players, ' Agnew said.
"If we show that commitment and intensity until the end of the season, we have a real chance of staying up".
Asked how to get the best out of Downing at a pre-Manchester United press conference, Agnew, who believes he is capable of keeping Middlesbrough afloat, told reporters: "Stewart's a Middlesbrough boy".
'It is a relief to get a goal but the ball has been bouncing, dropping, not quite going in.
Speaking before the game, Gibson had hinted Agnew could be Karanka's long-term successor as manager.
"There's more to him than people know and a bit more steel". One thing I know is how hard Aitor worked at this club, he's a tremendous guy and will go on to better things.
I have to be honest, it was a really hard day yesterday, a very emotional day, but then today the players have trained well and we look forward to the weekend.
His last job in England was with QPR back in 2015.