A group of fans paid for a plane to fly over the ground bearing a banner that read 'Ed Woodward A Specialist In Failure' before kick-off, while the visiting fans showed their support for the under-fire Mourinho by chanting his name for much of the contest.
The plane protest underlined the fact that a number of fans are siding with the manager following a tense transfer window in which several Mourinho targets were said to have been vetoed by Woodward.
When asked for his response to the banner, however, Mourinho offered an amusing response.
"I am not looking to the sky unless I am asking the guy for help!" he told a news conference. "I didn't see planes but Ed Woodward won 2-0 this afternoon."
United deserved to win what proved to be an eventful game on Sunday, with Paul Pogba missing a penalty and Marcus Rashford earning a straight red card in the space of two second-half minutes.
Burnley battled for a way back into the match but could not avoid a third straight league defeat, which came at the end of a week when they knocked out of the Europa League play-offs by Olympiacos.
"Disappointment is quickly thought about in the same process as the reality," manager Sean Dyche reflected to BBC Sport. "Manchester United are a very good side. They started hard and fast with the ball and defended strong.
"Sometimes there is balance to the disappointment. We got about them; the goal on half-time is a sickener for us. I thought Lukaku was excellent. It's a strange one because we come out of a useful period for experience but a tough period."