The Red Devils created more chances in the first half, with Martial and Mata both having good opportunities to give their side the lead, but mishitting their shots.
James came closest when his header was tipped over the crossbar by Pope.
But it was Burnley who got the breakthrough when Mee flicked on a free-kick to Wood who sent his shot past De Gea at the near post.
The visitors resumed second half with the same momentum as Rodriguez plays a one-two with Wood and his first touch just takes him away from Maguire. It looks like the angle was too tight, but he fired it over De Gea, off the crossbar and into the back of the net.
Ole Gunnar Solskjaer was hoping his side could get back to winning ways today as they try to close the gap to the top four but failed as Burnley claimed all 3 points.
The win sees Burnley move seven points clear of the relegation zone and they move up into 13th, just four points behind United.