Ben Chilwell grabbed the opener against his former club to set the tone for the win and further ease the pressure on manager Potter.
Kai Havertz's fine finish restored the visitors' lead after Patson Daka levelled and Mateo Kovacic's volley sealed it.
Chelsea forward Joao Felix hit the post and had a goal ruled out by VAR while Kiernan Dewsbury-Hall missed a sitter for the hosts before Wout Faes was sent off late on.
A third straight victory, after reaching the Champions League quarter-finals having knocked out Borussia Dortmund on Wednesday, should represent the start of the recovery under Potter.
It was just the second time Chelsea had scored three times in a Premier League game this season and there was a fluidity and clinical edge which has been lacking this term.
For Leicester, defeat edged them closer to the relegation zone having looked good to stay clear before the World Cup break.
Wins for Bournemouth and Everton left the Foxes just a point above the drop zone.
Victory for West Ham over Aston Villa on Sunday would leave Brendan Rodgers' men, who face trips to Brentford and Crystal Palace next, 17th after five straight defeats in all competitions.
Match of the Day presenter Lineker came to support his hometown club after the BBC told him to step back from hosting the highlights programme on Saturday in a row over impartiality.
Even James Maddison, the Foxes' brightest spark, could not inspire them and if Lineker had hoped for a relaxing afternoon watching his team he was mistaken as the hosts fell behind after 11 minutes.
Chilwell's corner was only half-cleared and recycled by Kalidou Koulibaly who tossed in a high cross beyond the far post where Chilwell waited having moved into the far side of the box.
There was plenty to do as the hanging ball dropped but the defenders controlled volley caught out Danny Ward at his near post.
The Foxes nearly hit back immediately but Daniel Amartey headed Maddison's vicious free-kick wide.
But Chelsea, led the combative Kovacic, were fighting and should have doubled their lead after 24 minutes.
Havertz spotted Felix's run and his perfect pass sent the striker clear but the Atletico Madrid loanee's chip over Ward bounced off the post.
Just 90 seconds later it was Leicester's turn to hit the woodwork when Dewsbury-Halls 20-yard effort clipped Koulibaly and then the bar.
The chances kept coming and Chelsea thought they had a second when Felix neatly finished a slick move involving Mykhailo Mudryk and Ruben Loftus-Cheek only for VAR to intervene and rule it out.
Felix should have been more disciplined, rather than straying offside, and he was then guilty of gifting Leicester a 39th-minute leveller.
The striker dithered on the edge of his own box and was robbed by Ricardo Pereira with Daka picking up the pieces to drive into the bottom corner from 20 yards.
Having survived two let-offs Leicester were encouraged and Maddison tested Kepa Arrizabalaga before the goalkeeper beat away Kelechi Iheanacho's snapshot.
But, just as the Foxes thought they had gained an edge, they fell behind seconds before half-time.
Enzo Fernandez's cute chip found Havertz and the forward lifted a gorgeous first-time lob over the stranded Ward from 12 yards.
Havertz's muted celebrations suggested he thought he was offside but Pereira had played him on meaning Chelsea had scored twice in the league for the first time since December.
They went for a third straight after the restart and Ward tipped Wesley Fofana's header over.
Chelsea were comfortable and, for the most part, kept Leicester at a distance, but Conor Gallagher - on for Felix at half-time - needed to be alert to block Harry Souttar's goalbound effort just after the hour.
There was a sense Chelsea needed a third and Havertz was denied by Ward before Dewsbury-Hall missed an incredible chance.
Kepa could only direct a cross to Harvey Barnes and the winger's header fell for Dewsbury-Hall six yards out with the goal at his mercy. But the midfielder completely miscued his shot and it rolled tamely into Kepa's arms.
Mudryk had a goal disallowed for offside after breaking through but a third finally arrived for the visitors with 13 minutes left.
Havertz outpaced Faes to cross for Mudryk and his header was met by Kovacic's crashing volley to wrap up the points.
It then got worse for Faes, who was sent off with four minutes left after second yellow card for a reckless challenge on Carney Chukwuemeka.