Eden Hazard hit the crossbar in the first half and Vardy punished Chelsea's profligacy six minutes into the second half to ease the pressure on Foxes boss Puel with Leicester's first win at Stamford Bridge in 18 years.
Maurizio Sarri's Blues, for whom Marcos Alonso hit a post in stoppage time, have now lost three of their last six Premier League games and desperately need a goalscorer to ease the burden on Hazard.
The Foxes had won just two of their nine games since the October 27 helicopter crash which resulted in the death of five people, including chairman Vichai Srivaddhanaprabha.
A run of two wins in 12 games had led to speculation Puel's position was in jeopardy, but a first win at Chelsea since September 2000 should ease the pressure. Re-live the live action below:
A day jammed full of festive football lays ahead. Eight games to fill you up on the final Saturday before Christmas, including Ole Gunnar Solskjaer's first game as interim manager of Manchester United.
But we start at the Emirates Stadium, for a lunchtime clash between Arsenal and Burnley - a game which has become rather important for the hosts...
Indeed, Unai Emery's men have been brought back down to earth after their long unbeaten run came to an end last weekend. First, Southampton shocked them in the league and then Arsenal tumbled out of the Carabao Cup in tame fashion, a 2-0 defeat to their closest rivals here on Wednesday.
So what team has Unai Emery picked as his side look to get back to winning ways?
Sean Dacha's Burnley battled hard against Tottenham last week and looked to have earned at least a point. But Christian Erisksen's late winner meant Dyche's side left Wembley empty-handed, and Southampton's win against Arsenal leave Burnley in the bottom three.
Nevertheless, Dyche has kept faith in the three-at-the-back formation introduced last time out, making two changes: Aaron Lennon has had a knee operation this week after suffering injury against his former club, while Robbie Brady also misses out with a knock. Chris Wood and Jeff Hendrick come in.
So a return for Mesut Ozil, though doubts remain over his role in Unai Emery's Arsenal. His exclusion in the week was another signal that his future is not entirely secure at the club.
However, as Ed Malyon contemplates, there may not be a natural fit for Ozil elsewhere in Europe:
A potential debutant on the Burnley bench today in young left-back Ali Koiki, who impressed as a trialist at Grimsby in the summer. He was spotted by Burnley playing in Croydon after a stint in Crystal Palace's academy. A debut at the Emirates today would be a rather nice Christmas present for the teenager.
Arsenal start with a four across the defence. There was some chatter that Mohamed Elneny may have started at centre-back, but the Egyptian is alongside Matteo Guendouzi and Grait Xhaka in midfield.
Joe Hart is forced into an outstanding save low to his right as Ainsley Maitland-Niles finds space down the right, while James Tarkowski does well to react quickly and clear with Arsenal players ready to turn the rebound home.
Burnley break and Ashley Westwood is freed by Ashley Barnes breaking through the middle, but the midfielder's finish is scuffed somewhat, flicking off Bernd Leno and earning Burnley an early corner.
A quick start to this game. Chris Wood is left alone by the Arsenal defence (believing him to be offside) and the home side are fortunate that Wood has to come back to the ball and thus unable to run away from them.
Mohamed Elneny miscontrols towards his own goal, allowing Jack Cork to nip in and surge beyond Elneny.
Cork rather wastes a good opportunity to force a chance, turning down both the shot and the pass and running himself into a group of Arsenal defenders, and Elneny eventually works back to dispossess him.
Alexandre Lacazette does well to find space on the left and the cross comes across to the far post.
Mohamed Elneny leaps highest and heads powerfully towards the far corner, but there are enough Burnley bodies there to block and scramble the ball away after a couple of ricochets.
Burnley settle things down for the first time in this game's rather hectic infancy, showing competency in stroking the ball between midfield and defence.