Firmino and Mane each scored again in the second period, the Senegalese finally killing Burnley off after Johan Berg Gudmundsson set up a nervy finish in stoppage time.
The wind was swirling around Anfield and played a role in Burnley's sixth-minute goal as Westwood's deep corner swung over Alisson Becker and inside the far post, although the Liverpool goalkeeper was impeded by James Tarkowski.
Jurgen Klopp's side were briefly flummoxed by the bizarre nature of the opening goal but soon began to create chances, with Firmino cutting inside and getting off a shot which looped narrowly over the crossbar via a Tarkowski block.
The Brazilian was not to be denied on 19 minutes when a one-yard tap-in ended his six-game barren run. Mohamed Salah exchanged a one-two with Georginio Wijnaldum by the right byline and his cross somehow squeezed through Tom Heaton and Tarkowski to leave Firmino with the simplest of finishes.
Adam Lallana was a surprise inclusion in the Liverpool lineup for only his fourth Premier League start of the season, with Klopp seemingly seeking more creativity in his three-man midfield following three blanks in four matches in all competitions.
However, it was the 30-year-old's industry which earned the Reds their second goal as he charged down a clearance from Phil Bardsley which would have fallen for Salah if not for a last-ditch tackle from Charlie Taylor. The ball rolled into the path of Mane, who curled past Heaton with aplomb on 29 minutes.
The Burnley front partnership of Ashley Barnes and Chris Wood have combined for 20 goals this campaign, but the visitors could not provide any service to the two target men, who were unable to register a single attempt on goal between them.
Dwight McNeil was Burnley's brightest attacking spark and he got the better of Trent Alexander-Arnold on several occasions during a period of pressure from the Clarets at the start of the second half, but his good play did not yield any clear chances.
At the other end, there was a golden opportunity for Wijnaldum on 66 minutes as he headed Andrew Robertson's cross wide.
On a weekend of comebacks in the Premier League, Liverpool would have been wary of their vulnerable one-goal cushion before Firmino gave them breathing space with a 68th-minute third goal, to which Burnley again contributed to their own downfall.
Heaton's miscued clearance went straight to the feet of Salah who ran down on goal, only to be denied just before pulling the trigger by another perfectly-timed tackle from Taylor, who was unfortunate that the ball fell kindly to Firmino six yards out.
Burnley made Anfield nervy again by pulling one back on 91 minutes, two of their substitutes combining as Matej Vydra picked out Gudmundsson with a measured square pass which the Icelandic international swept past Alisson at his near post.
However, Liverpool quickly restored their two-goal margin as Daniel Sturridge sent Mane through on goal, and the winger stayed on his feet under the challenge of Heaton to round the Burnley goalkeeper and slot home his 16th goal of the season.
Liverpool are now one point behind Man City and will be above them by the time the champions next play in the Premier League, if they can beat Fulham. Meanwhile, Burnley are only two points above the relegation zone after a third consecutive defeat.
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold (Sturridge 86'), Matip, Van Dijk, Robertson; Fabinho, Wijnaldum (Henderson 68'), Lallana (Keita 77'); Mane, Firmino, Salah
BURNLEY (4-4-2): Heaton; Bardsley, Tarkowski, Mee, Taylor; Hendrick (Gudmundsson 79'), Cork, Westwood, McNeil; Barnes (Vydra 86'), Wood (Crouch 79') body check tags ::