It looked as though City were on their way to a first Championship defeat since December 7 until Drinkwater struck in the fourth minute of added time.
Two goals from top scorer Danny Ings set the hosts on their way as Burnley moved into the automatic promotion places at the expense of QPR, who play on Monday.
Millwall opened the scoring through Martyn Woolford on 27 minutes but the lead lasted just two minutes before Ings equalised.
Burnley went in at the interval ahead as Dean Marney struck just before half-time and Ings made sure of the win with his 24th of the season just after the hour mark.
Brian McDermott's return to the dugout saw Leeds come from behind to earn a 2-1 victory over Yeovil at Huish Park.
McDermott had been advised to stay away from Elland Road last weekend in the midst of the controversial takeover by Massimo Cellino, having been told his time at the club was over.
But having been reinstated on Monday as Cellino waits for Football League approval for his takeover, McDermott saw his side emerge on top in a match played in testing conditions.
Ishmael Miller opened the scoring for the hosts but then turned villain when he blazed a penalty over that would have made it 2-0.
Ross McCormack, fresh from his hat-trick against Huddersfield last week, levelled with a wonderful curling effort to draw them level seconds after the break before a 40-yard free-kick from Stephen Warnock caught the wind and looped over Yeovil goalkeeper Marek Stech.
In-form Sheffield Wednesday extended their unbeaten run to 11 matches with a comfortable 2-0 win against 10-man Reading at the Madejski Stadium.
Reading centre-back Alex Pearce was sent off in only the ninth minute after conceding a penalty. Chris Maguire slotted home the spot-kick and Benik Afobe added a second goal from close range in the second half.
Blackpool debutant Andy Keogh scored a late equaliser as the hosts claimed a 1-1 draw at home to Nottingham Forest, who took the lead through Jamaal Lascelles.
David McGoldrick hit his 16th goal of the season as Ipswich bagged two late goals to snatch a 2-2 at Barnsley.
McGoldrick's 84th-minute free-kick and Christophe Berra's close-range finish three minutes earlier cancelled out the Tykes' two-goal lead, given to them by Chris O'Grady and Tomasz Cywka.
Bolton's struggles near the foot of the standings continued as Dougie Freedman's men threw away a two-goal lead to draw 2-2 against Bournemouth.
First-half strikes from Lee Chung-yong and Lukas Jutkiewicz had seen the home side look in complete control until a long-range Lewis Grabban goal saw the Cherries back in the game at the break. Simon Francis then headed in a scrappy second-half equaliser.
A double from on-loan Manchester United striker Federico Macheda gave Birmingham a 2-0 victory over fellow strugglers Charlton, while Huddersfield ground out a 1-0 win over Wigan thanks to Adam Clayton's lone strike.
Middlesbrough and Blackburn drew 0-0, while Leonardo Ulloa's goal saw Brighton to a 1-0 win at home to Doncaster, who also lost striker Billy Sharp to a late red card.
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- Danny Ings