Substitute Jesse Lingard saved Manchester United while it was a record-breaking outing for Tottenham's Harry Kane on a busy day of English Premier League action.
Champions Chelsea beat Brighton 2-0 at Stamford Bridge.
Roberto Firmino scored twice as Liverpool eased to a 5-0 win over bottom club Swansea City to remain fourth.
Lingard struck twice in the second half as Manchester United fought back from 2-0 down to earn a 2-2 draw with Burnley at Old Trafford.
Burnley grabbed a surprise third-minute lead when, after a goalmouth scramble, Ashley Barnes fired home from close range and it could have been 2-0 when Scott Arfield's volley clipped the top of the bar.
All Whites striker Chris Wood didn't play for Burnley because of injury.
United are now 12 points behind Manchester City who play tomorrow.
United piled on plenty of pressure but Burnley doubled their lead in the 36th minute thanks to a superb curling free kick from 30 yards by Belgian midfielder Steven Defour which beat the diving David De Gea.
United manager Jose Mourinho made a double substitution at the interval and Lingard got the home side back in the game with a clever back-heel flick in the 53rd minute.
Mourinho's side totally dominated the second half but were frustrated by a sturdy Burnley defence until Lingard drove home the equaliser in stoppage time.
Earlier, Kane rounded off a sensational 2017 by scoring a record number of Premier League goals and hat-tricks in a calendar year, as well as toppling Lionel Messi as the top scorer in Europe as a rampant Tottenham thrashed Southampton 5-2.
The England striker followed up his treble over Burnley on Saturday by heading in a Christian Eriksen free-kick in the 22nd minute for his 37th league goal of 2017, beating Alan Shearer's 22-year-old record for the most goals in England's top flight in a calendar year.
Kane then overtook Messi as the top scorer of the year in Europe for club and country by tapping in Son Heung-min's low cross in the 39th minute. In the second half Kane scored his side's fifth goal of the game to become the first player to hit six Premier League hat-tricks in one year.
"It was hard not to think about it going into the game being level," he told BT Sport.
"As always I wanted to win the game first and foremost but of course I wanted to score as a striker. To get that goal early on and get that record was a great feeling and I could enjoy the rest of the game."
Elsewhere, an own goal by Leicester keeper Kasper Schmeichel gifted Watford a 2-1 home win over the Foxes while West Brom's winless run continued in a goalless stalemate with Everton.
Huddersfield and Stoke drew 1-1 and Bournemouth and West Ham also drew 3-3.