First-half goals from Argentina duo Giovani Lo Celso and Erik Lamela saw Spurs through to a round-four meeting with Southampton, although some wasteful finishing and a late goal from Boro substitute George Saville set up a nervy finish.
Tanganga, who was handed a surprise Premier League debut on the right of a back three in Saturday's 1-0 defeat to Liverpool, was a dynamic presence at right-back and almost got on the scoresheet this time out.
"I trust him. I know the way he can read the game and learn," Mourinho said afterwards, as quoted by BBC Sport on Tuesday.
"He absorbs all the information. He is not that kind of modern kid who is worried with pathways, the pathways it to work hard and take the opportunity which he did.
"Now people know how good he is."
Tottenham were unable to close out a clean sheet – they have still only managed one during Mourinho's 14-game tenure – and their head coach felt a failure to kill the game off led to needless complications.
"We were close to 3-0 so many times," he said. "I told my players at half-time, if we don't score 3-0 then it could be 2-1 and we would be in trouble. And it happened.
"We knew the opponents were hard. They brought on [Rudy] Gestede and went direct and made problems and when it was 2-1 we knew it was going to be difficult.
"We tried our best, the boys are trying their best. They dealt well with many set-piece situations.
"We did lots of things well, we conceded the goal, a bit frustrating, but more frustrating was that we did not score three, four or five."
After snapping a four-match winless run, Spurs travel to in-form Watford in the Premier League this weekend.