Hart performed well in the defeat to United last time out, making a number of key saves including keeping out Paul Pogba's second half penalty, but ultimately it couldn't help Burnley achieve a result.
Asked about the banner, which flew as his team were warming up and he was in the tunnel, Mourinho said: "I didn't see planes".
"They say I'm in danger, but I don't think so", Mourinho told Gazzetta dello Sport and La Repubblica.
"I think we started winning this match because of Old Trafford, after Spurs".
Mourinho will be aiming for his Manchester United side to establish themselves as one of the main contenders for the title this season.
But what had been an unqualified United success to that point was somewhat diminished when Rashford headbutted Bardsley after being cynically fouled by the 33-year-old former United defender.
"(Referee Jonathan) Moss had a good game, a good performance, really.
The team sheet saw a few changes from the defeat against Spurs, with the formation going back to a 433 and Victor Lindelöf and Marouane Fellaini coming into the team replacing the injured Phil Jones and Fred.
United next travel to Watford in the league on Sep 15.
The media officer relented and when Mourinho had finished eulogising over the Belgian, he went back to the question she wanted to take from an Italian journalist who, it transpired, wanted to talk about United's Champions League opponents Juventus. "This two week period now gives us a chance to rest and work, but certainly refresh ourselves for the challenge which is the Premier League".
"The break comes at a good time for us; it's as simple as that", said the Turf Moor goalkeeper.
Mourinho claimed that Pogba will remain United's penalty taker, despite his miss, should he wish to remain on duty.
At Old Trafford we lost at home but the fans were unbelievable. United's best course of action might actually be to sack both Mourinho and Woodward but as long as United's stock keeps climbing and they keep making money, the Glazer's are keeping Woodward.
"Many of the problems stem from Ed Woodward's continuous undermining of Manchester United managers in the transfer market, in matters of the playing squad, and in his inconsistent approach to recruitment and retention of managers and players".