Premier League RESULTS: Chelsea edge Crystal Palace in five-goal thriller as Welbeck's overhead kick gives Watford huge boost and condemns Norwich
In a game full of screamers and controversy, Palace refused to lie down despite finding themselves two-goals down twice.
The Blues went in front in contentious fashion when Willian played on despite former Chelsea player Gary Cahill pulling up with a hamstring injury.
Although terribly unfortunate for Palace, most pundits were agreed the Brazilian was right to do so, and Olivier Giroud made no mistake with the cutback.
Chelsea appeared out of sight when Christian Pulisic netted a screamer to continue his bright restart, but then Wilfried Zaha made the American's strike look ordinary.
The Ivorian scored his first Premier League goal in 16 matches with a thunderbolt from 30 yards - though questions will linger over Blues goalkeeper Kepa Arrizabalaga.
Palace were in the ascendancy for much of the second-half, but Chelsea caught them on the break to restore their two-goal cushion through substitute Tammy Abraham.
But the hosts were not done yet - and made it game on once again by going up the other end immediately to half the deficit through Christian Benteke.
Palace hit the post and Kurt Zouma made an astonishing last-ditch challenge, but Chelsea held their nerve to get through the last ten minutes, and they will stay third tonight unless Leicester beat Arsenal.
At the other end of the table, Watford took a massive stride of their own, but for them it was towards safety with a 2-1 victory over Norwich.
In a fast start, Emi Buendia put the visitors ahead on four minutes before Craig Dawson cancelled it out six-minutes later.
And then Danny Welbeck stole the show with a stunning overhead kick, which might just have condemned Norwich to next season in the Championship while giving Watford a massive survival boost.
Chelsea hung on for dear life and took a huge stride towards Champions League qualification with a thrilling 3-2 victory over Crystal Palace on Tuesday.