Diego Costa scored the winner as Chelsea secured a fine comeback win at Liverpool and there was more good news for Jose Mourinho as Manchester City could only pick up a point at struggling Queens Park Rangers.
But Chelsea remain four points clear at the top of the table after Southampton, their nearest challengers, won at home to Leicester City thanks to Shane Long's double.
Manchester United endured a frustrating afternoon at home to Crystal Palace before substitute Juan Mata scored the only goal of the game in the 67th minute, while Burnley finally picked up their first three points of the season as Ashley Barnes' goal beat Hull City.
West Ham United and Aston Villa played out a goalless draw in the afternoon's other game.
Costa's 10th goal of the season maintained Chelsea's unbeaten record and inflicted yet more pain on Liverpool at Anfield.
Back in April their win on Merseyside signalled the beginning of the end of the Reds' title dream and while this 2-1 victory was not as heartbreaking or damaging for the home side it was just as frustrating.
Midfielder Emre Can's first goal since his summer move from Bayer Leverkusen -- his shot deflected in off Gary Cahill -- had given Liverpool an early lead but it has not been a situation they have been comfortable with all season.
Cahill's scrambled 13th-minute effort via the goal decision system ensured Chelsea manager Jose Mourinho could have no complaints and he would have been positively overjoyed with Costa's clinical first-time effort midway through the second half.
Rodgers wanted a penalty late in the game for handball in the area against Cahill, who had an eventful afternoon, but referee Anthony Taylor waved away the claims.
Champions Manchester City now trail Chelsea by eight points after they drew 2-2 at QPR.
Charlie Austin had the ball in the net three times in the space of a matter of minutes early on for the home side but only one of his efforts counted. The first was rightly ruled out for offside before the striker capitalised on a poor free-kick in his own box by goalkeeper Joe Hart to net from the edge of the area. But referee Mike Dean spotted that Hart accidentally kicked the ball twice and the laws of the game state that Hart must retake.
Austin finally did break the deadlock when he burst through and shot under Hart. The lead did not last long as Sergio Aguero got away with controlling the ball in the box with his hand before levelling.
QPR again took the lead after the break as Martin Demichelis headed the ball into his own net, but Manuel Pellegrini's blushes were spared to some extent when Aguero's fine goal ensured the game would finish all-square.
Shane Long brightened up a dank afternoon at St Mary's, netting a brace after coming off the bench to continue Southampton's best-ever start to a top-flight season with a 2-0 win over Leicester City.
Eyebrows were raised when Saints forked out 12 million pounds to bring in the Republic of Ireland international from Hull during a well-documented summer of upheaval on the south coast.
On this wet afternoon, though, Long followed in his club's footsteps by confounding the doubters, scoring twice shortly after coming on as second-half replacement to secure the points -- their fifth successive victory in all competitions -- and stay in touch with Chelsea.
Mata stepped off the bench to score the winner as Manchester United returned to winning ways despite a lacklustre display against Crystal Palace.
Just four minutes after coming on for Adnan Januzaj, Mata curled a deflected shot past Julian Speroni to hand Louis van Gaal's men a 1-0 win.
The result was a welcome one for United fans and their manager, who had not won any of his previous three matches.
But in truth, he will still feel a bit of the lousiness he spoke about at his prematch news conference on Friday.
Burnley finally claimed their first win of the season as Ashley Barnes' header steered them to a 1-0 victory over Hull City at Turf Moor.
In what was their 11th Premier League game of the campaign, the decisive moment came for the Clarets in the 50th minute when Barnes connected with Kieran Trippier's cross.
They might have scored earlier, with Danny Ings shooting narrowly wide before being denied by a Steve Harper parry, the loose ball then being sent back towards his own goal by Paul McShane and cleared off the line by Robbie Brady.
They were then playing against 10 men for the final 20 minutes -- Hull captain Curtis Davies having been forced off with an injury when his side had already made three substitutions -- but Sean Dyche's team could not add to their tally.
Aston Villa ended their losing streak after battling to a 0-0 draw at West Ham United.
Paul Lambert's side had suffered six straight defeats since winning at Liverpool in mid-September to plummet down the table.
However, Villa produced a determined, defensive display at the Boleyn Ground, where goalkeeper Brad Guzan denied Andy Carroll -- back from an ankle injury -- with a brilliant save in stoppage time to frustrate the Hammers, who had been looking to move back up into the top four following their impressive run of form.