Mourinho was sent to the stands at half-time for arguing with referee Mark Clattenburg and Herrera went on 68 minutes for two bookable offences against Dean Marney.
United were denied victory by an outstanding display by Tom Heaton and the Burnley goalkeeper produced a strong of fine stops.
The pick was when he stuck out his arms and legs to deny Zlatan Ibrahimovic with a stop reminiscent of Peter Schmeichel.
Juan Mata, captain for the first time under Mourinho, hit the post and Ibrahimovic looped a header onto the bar as Burnley held out to earn their first away point of the campaign.
1) Mata’s stock has never been higher at United
After looking like he might be heading for the exit when Jose Mourinho took over, he is revelling in the post-Rooney world at Old Trafford.
Not only has he taken Rooney’s favoured No 10 role, he captained United for the first time under Mourinho in a sign of how highly the Portuguese rates him after letting him quit Chelsea.
2) It looks a long way back for Mkhitaryan
Mkhitaryan did not even make the bench, even though Mourinho backed him yesterday.
The Armenian playmaker, signed for £26million from Borussia Dortmund, is not injured, but Mourinho says he lacks match fitness.
Yet he does not feel he is ready to add to his meagre tally of 104 minutes of Premier League action. A percent Catch 22-situation for Mkhitaryan.
3) Heaton deserves a real chance with England
Heaton’s Three Lions career consists of three minutes against Australia so far in a friendly and he showed against Manchester United that he is ready for the big stage.
The Burnley keeper was superb in front of the watching England No 2 Sammy Lee and the pick of his saves was a point-blank point from Zlatan Ibrahimovic, which was reminiscent of Peter Schmeichel.
4) Mourinho is in more trouble with the FA
Mourinho was sent off at half-time for arguing with referee Mark Clattenburg.
He cheekily tried to sit beside the home dug-out in a fan’s seat before fourth official Kevin Friend dispatched him to the directors’ box from where he passed notes via his staff to assistant Rui Faria.
Mourinho is already facing a misconduct charge for commenting about referee Anthony Taylor before the Liverpool game and could be hit with a touchline ban.
5) Gray looked rusty
Gray was back for Burnley after serving his four-match ban for his homophobic tweet.
The striker had not played for six weeks and it showed in his display.
He had one great chance early in the second half when he broke from just inside his own half, but was tackled by Luke Shaw. He was replaced by Ashley Barnes just before the hour mark.