The Red Devils were held to a second 2-2 draw in a row in the Premier League despite Jesse Lingard's double at least saving them from a defeat at Old Trafford.
Ashley Barnes gave the Clarets the lead after three minutes and they doubled their advantage through Steven Defour's fine free-kick, before Lingard rescued a point with a clever flick and an injury-time equaliser.
Mourinho blasted his players for "childish" finishing in the 2-2 draw with Leicester City last week, in which they spurned a host of chances before conceding at the death, and he was again frustrated not to see United make the most of their dominance.
He told BBC Sport: "We should have won, yes. I take satisfaction by the way the team played, by the way the team reacted, to try to recover from 2-0 down.
"The boys fought against the 2-0 at half-time, so nothing at all negative to say about my players. I praise their spirit, their reaction, the way they accept all the risks we took in the second half.
"I don't know how they score a goal from a free-kick, rebound here, rebound there, a similar goal that we conceded in other matches and then they defend for 95 minutes. Second shot, they score again.
"For us to score two goals we need to play in their half for 45 minutes, with only one defender. The end product of so much dominance, two goals, is a miserable number."
Mourinho, who complained last week about United's December schedule, denied the fixture pile-up was to blame for them dropping more points but also refused to accept they are going through a difficult spell.
"It was the same for us and the same for them, they are also one of the teams that is punished by the fixtures," he said.
Asked if they are in a rough patch, he replied: "Why? Because we deserve to win matches? Because the opponent went to our goal twice and they score twice? The quality of our play, the risk that we took and also against Leicester, the team that deserved to win both matches is us."
Burnley boss Sean Dyche was proud of the way his side defended but was unhappy with the award of a free-kick against Sam Vokes that led to Lingard's leveller.
6 - No side has won more points away to the Premier League's 'big six' this season than Burnley. Crackers. pic.twitter.com/8ynOVZSvvG— OptaJoe (@OptaJoe) December 26, 2017
"I have a lot of pride in a team that is stretched at the moment," he said. "The game plan is you're going to have to defend and do all of the ugly details of the game well.
"We scored early, we didn't relax on that, we scored again. Second half, they dominated a lot more. To come out with a point is pleasing.
"My only frustration is the final decision on Sam Vokes, I don't know what he's supposed to do. I look at their team play and their bench they don't look too vulnerable."