Elsewhere, Southampton humble champions Leicester
While all around them faltered to some degree or another, Arsenal provided drama late on and kept the pressure on leaders Chelsea.
Alexis Sanchez's 98th minute panenka penalty was enough to earn them all three points, despite conceding a penalty themselves in stoppage time.
Chelsea, meanwhile, made it eight straight wins at home with a 2-0 victory over Hull City at Stamford Bridge.
Earlier in the day, Leicester City were beaten by out-of-form Southampton at St Mary's. James Ward-Prowse, Jay Rodriguez and Dusan Tadic did the damage against the champions, who haven't won in the league this year.
Southampton inflicted further pain on defending Premier League champions Leicester with a 3-0 drubbing at St. Mary's.
The home side dominated from the opening whistle and eventually James Ward-Prowse opened the scoring in the 26th minute when his shot went past Kasper Schmeichal into the far corner of the Leicester net.
Jay Rodriguez doubled the lead from close range in the 39th minute when Leicester failed to clear a cross into their box.
The Saints went into the break two goals to the good as Leicester rarely threatened Fraser Forster.
Southampton picked up from where they left off after the interval and the match turned into shooting practice for Claude Puel's side.
Wes Morgan was lucky to have an own goal chalked off for offside but he then gave away a penalty in the dying minutes of the match.
Dusan Tadic blasted the ball into the roof of the net to round off a comfortable 3-0 win.
Today threatened to be a disappointing day for Arsene Wenger. His side went ahead through Shkodran Mustafi and looked as though they were headed towards back-to-back league wins.
Granit Xhaka's second half sending off, however, was not in the script. Arsenal did not crumble, but with 10 men on the field, Burnley inevitably grew back into the game.
Supporters grew anxious and their fears were proven well-founded in stoppage time when Jon Moss awarded the visitors a penalty. Andre Gray stepped up in the third minute of stoppage time. Straight down the middle. Level.
Wenger lost his cool on the sideline and vented his frustration at Anthony Taylor, even pushing him when he was sent to the tunnel.
One man who didn't lose his cool was Alexis Sanchez. With seven minutes of additional time at the end of the game, there was still time. Moss would point to the spot once more and award Arsenal a penalty in the very last minute of the game. Up steps the Chilean.
In the last minute of the game, Sanchez had the audacity to execute a panenka penalty to win the game for Arsenal. Often, these are the sort of games that would derail Arsenal during a tilt at the title. Throwing away points late on in games tends to ask questions of Wenger's men.
Today's late victory will provide a huge boost to the team and serve to motivate them ahead of a tight run-in, at a time where they have struggled in recent years.
Alexis Sanchez applies a panenka finish in the 98th minute to earn Arsenal victory at home to Burnley. Image: Nigel French EMPICS Sport
Antonio Conte watched his Chelsea side extend their incredible home form with a 2-0 victory over Hull City.
Chelsea made it eight wins on the bounce at Stamford Bridge, winning 10 of their 11 home games this season. They've conceded just six goals in that time and there only defeat came with the visit of Liverpool.
The Londoners were back to winning ways last week with victory over Leicester City and now looked to have resumed normal service after Tottenham halted their winning run.
The Italian will be aware of the importance of intimidating teams who visit the Bridge and with the visit of title rivals Arsenal at the start of next month, Chelsea will have a physiological edge heading into the game.