Stand-in Clarets skipper James Tarkowski, who took the armband in the absence of the injured Ben Mee, gave the hosts the lead on the stroke of half-time.
The defender poked the ball past Dean Henderson from close range after Jay Rodriguez had flicked on Dwight McNeil's set-piece.
John Egan, who moved to Brentford following Tarkowski's departure in 2016, levelled with 10 minutes remaining when firing the ball into the top corner.
Here are the player ratings...
1. Nick Pope 7
Couldn't quite add a 14th clean sheet of the campaign, but there was very little he could do to keep out Egan's equaliser. Saved well with his feet to deny McBurnie early on and his hyandling was very good once again.
2. Phil Bardsley 6
Didn't get a moment to draw breath in the first half when Stevens, Robinson and Osborn looked to affect the play. Tested Henderson with a stinging first half drive and enjoyed a little bit more protection from Pieters after the break.
3. James Tarkowski 7
Took the armband in the absence of Mee and delivered when giving Burnley the lead at the break. Didn't lose a header, strong off the ball, but stayed deep from United's short corner to play goalscorer Egan onside.
4. Kevin Long 6
Had big boots to fill with Mee sidelined through injury, but did everything that was asked of him. On his heels when caught out by McGoldrick early on and then settled into the game. Showed his strength and made a crucial block from Berge.