With Manchester City, Liverpool and Arsenal coming up in their next four games, Burnley were intent on putting their fellow relegation strugglers to the sword early on.
Jóhann Berg Guðmundsson’s eyes lit up when he found an opening inside the penalty area with just six minutes on the clock, but his finish lacked conviction and paved the way for Ross Barkley to step up for the Hatters and temporarily take the game by the scruff of the neck.
An impressive spell of Luton pressure saw the former England international ping a sequence of pinpoint 40-yard balls and force James Trafford into a smart save from long range, but the high-quality passing clinic failed to yield any meaningful chances and Burnley capitalised.
Just 10 minutes before the break, teenager Wilson Odobert burst down the left-hand side and crossed in for Zeki Amdouni, who fired into the net through a crowd of bodies to send Vincent Kompany haring down the touchline in celebration.
Despite the importance of the three points at stake, Burnley spent much of the second period pushing for a second rather than battening down the hatches.
Vitinho was the first to threaten with a well-struck effort from distance, before Odobert’s deflected strike forced Thomas Kaminski to scramble across his goal-line and palm to safety.
The Hatters’ substitutions initially failed to incite the kind of stranglehold they had on the contest for a large part of the first-half, but just as it appeared that all hope was lost, the visitors dramatically equalised in the 93rd-minute.
Alfie Doughty’s cross was headed home by Morris, and after a long VAR review to check assess a potential foul on Trafford, the goal was given to the delight of the away fans, who travel back down south with a vital point which draws them level with 17th-placed Everton.
Burnley, meanwhile, are four points adrift as they and Rob Edwards’ side head into a two-and-a-half-week break from PL action.
Flashscore Man of the Match: Wilson Odobert (Burnley)