England midfielder Alli had headed Spurs level, cancelling out a first-half own goal scored by Fernando Llorente - the striker deputising for the injured Harry Kane, who is out until March.
Alli could well be set to join Kane on the sidelines after limping off in the closing stages, but what looked like becoming a nightmare Sunday for Spurs at Craven Cottage was saved at the death by Winks.
Georges-Kevin Nkoudou, Alli's replacement, delivered a wonderful whipped ball into the box from the left wing and Winks was there to condemn struggling Fulham to another defeat.
Fulham scored two own goals in a 2-1 loss at Burnley last weekend, although one was later given to Jeff Hendrick, and the tables were turned when they took the lead in the 17th minute.
Jean Michael Seri's corner from the right wing did not look particularly dangerous but Llorente clumsily turned the ball into his own goal.
Llorente and Ryan Babel missed good headed chances and Fulham then had a second goal ruled out for offside when Aleksandar Mitrovic nodded home the rebound after Lloris had parried away a typically powerful Andre Schurrle drive.
Spurs improved after the break and they levelled in the 51st minute, Tim Ream's fluffed clearance allowing Christian Eriksen to float a cross to Alli at the back post for a simple header.
Both sides had penalty appeals rejected and Llorente wasted another chance by heading wide Danny Rose's cross, the left-back having earlier hit the crossbar with a deflected drive.
Alli hurt his left hamstring in the closing stages to add to Spurs' growing injury list, but an unlikely pair of heroes combined to create the winner deep into stoppage time.
Nkoudou's cross was inviting and Winks timed his run perfectly to nod in from close range.
Fulham survival hopes fading
Fulham were excellent in the first half and could have led by two or three goals at the break, but their defensive issues were highlighted after the interval. With 51 goals conceded in the Premier League, a seven-point gap to safety looks too big. Spurs, meanwhile, are back to within nine points of leaders Liverpool, but Champions League qualification may be a more realistic goal.
Alli comes up trumps for Spurs
Someone had to deliver for Spurs with Kane not fit to play and it was Alli who produced when his side needed a goal. The England midfielder hit 18 in the 2016-17 Premier League season and he may need to record a similar return with Kane set to miss at least 10 games. Spurs will hope Alli's late hamstring injury was not as serious as it looked.
Llorente lost in Kane's absence
Filling Kane's boots was always going to be a tough task for Llorente, restricted to six Premier League appearances as a substitute this term. But the striker looks a poor fit for a Spurs side who rely on mobility in their front line and the less said about his awful own goal the better.
