Substitute Daniel Sturridge hit the post with a low, left-footed drive from 20 metres and Senegal forward Mane was quickest to the rebound, sliding in to convert from close range.
The victory pushed Juergen Klopp's Liverpool side within six points of leaders Chelsea who won 1-0 at Crystal Palace on Saturday.
"We didn't give one chance away and that's very important against an opponent that only wants to disturb your game," said Klopp.
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"The plan of Everton was wild football. They were chasing us everywhere, it is really difficult to play against this ... it's a wonderful feeling to win."
In a match with few chances, Everton lost goalkeeper Maarten Stekelenburg because of a leg injury after an hour and understudy Joel Robles had to make a fine, diving save from Roberto Firmino with 10 minutes to go.
Stekelenburg had denied Firmino in the 50th minute as Liverpool dominated, palming the ball away while Nathaniel Clyne's effort from the rebound was blocked.
Firmino also attempted a scissors kick from a Clyne cross but his effort sailed over.
Tempers flared when Liverpool midfielder Jordan Henderson was caught on the ankle by a late tackle from Ross Barkley in the second half.
There was scuffling among players from both sides before Barkley and team mate Seamus Coleman were booked by referee Mike Dean.
Earlier, the first half turned into a stalemate with both keepers seriously under employed.
Divock Origi missed a chance in front of Everton's goal following a Clyne cross and Ramiro Funes Mori headed narrowly wide at the other end.
Liverpool then pushed back Everton, who have now gone 12 Merseyside derbies without a win and 20 with only one victory, in the second half and were ultimately worthy winners.
(Reporting by Rex Gowar; Editing by Tony Jimenez)
