United remain in contention for Champions League qualification after a hard-fought victory on Sunday, which ended with rival players separated by security staff in the tunnel when punches were reportedly thrown.
Middlesbrough captain Ben Gibson and United's former England midfielder Ashley Young were restrained by teammates as tempers flared.
Mourinho refused to comment, but Middlesbrough caretaker manager Steve Agnew insisted his players were reacting to provocation after a stoppage-time bust-up between United defender Eric Bailly and Rudy Gestede, which initially appeared to include the duo biting each other before replays showed it was only a heated clinch.
"I'm picking up there's something gone on in the tunnel," Agnew said.
"That many bodies were there, it was hard to tell with all the pushing and shoving.
"What triggered it off I don't know? I've got a group of very well behaved players so I can assure you it wouldn't be one of ours.
"What I would say it would have been a reaction because of the spirit they have."
Mourinho had enflamed matters before the game by suggesting he 'knew the names' of the players he claimed had been responsible for the departure of Aitor Karanka, his former Real Madrid assistant who was sacked as Middlesbrough boss last week.
The Portuguese headed down the tunnel at the final whistle without acknowledging his opposite number, though Agnew confirmed the pair had shaken hands in the tunnel afterwards.
"Was this justice for Karanka?" Mourinho said after seeing his side extend their 18-game unbeaten league run to end a 19-week stay in sixth by climbing to fifth, two points ahead of Arsenal and Everton.
"I think justice for Karanka is the work he did here, people can never delete that. He took the club form the brink of League One to the Premier League.
"I know what I'm saying, he could have left for a bigger club and yet he stayed then in the end he loses his job.
"Despite all that's happened, I still think he'll be unhappy we have won, because he wants this club to stay in the Premier League."
Middlesbrough's hopes of achieving that aim look increasingly slim after an 11-game three-month run without a league win.
They at least ended their lengthy goal drought as Gestede pounced on a mistake by Chris Smalling to halve the arrears.
Hopes of salvaging an unlikely draw to chip away at the five-point gap to safety faced by a side currently second from bottom were extinguished by Victor Valdes' stoppage-time slip which allowed Antonio Valencia to walk the ball into an unguarded net.
Goals either side of half-time from Marouane Fellaini and Jesse Lingard had put the visitors in control.
"It's a big win that keeps us alive in the competition for fourth place," Mourinho added.
Given their presence in the Europa League, United retain two potential avenues to next season's Champions League, and Mourinho added: "Do I prefer to qualify for the Champions League by finishing fourth or winning Europa League? I prefer Europa League. There's a prestige to winning a trophy."
"To push Manchester United all the way is something I was really pleased with," added Agnew, who has had no assurances as to the length of his temporary tenure beyond Middlesbrough's next game at Swansea at the start of next month.
"It's a display that gives us hope for the future and I've seen enough of these players to know we're more than capable of winning games."