Senegalese flier Mane netted twice in three first-half minutes as Jurgen Klopp's Liverpool claimed a first league win of 2017 at the sixth attempt and ended Spurs' nine-game unbeaten run.
Liverpool moved back into the top four, above Manchester City, who visit Bournemouth on Monday, and Manchester United, who briefly climbed above them after beating Watford 2-0.
Tottenham remain second, but they are now level on points with Arsenal after Alexis Sanchez scored twice to earn Arsene Wenger's side a much-needed 2-0 victory over Hull City.
Spurs' setback means Chelsea can move 12 points clear at the summit if they win at Burnley on Sunday.
While the title seems destined for Stamford Bridge, the battle for positions below Chelsea could not be more tight, with two points separating Spurs in second and United in sixth.
Liverpool badly missed Mane during his time at the Africa Cup of Nations with Senegal and he gave a dazzling display of his qualities with two goals in the first 18 minutes at Anfield.
For the first goal, in the 16th minute, he raced onto Georginio Wijnaldum's pass before lifting a shot past Hugo Lloris.
Two minutes later he made it 2-0 with his 11th goal of the season, sparking a raid on the Spurs box by robbing Eric Dier and then slamming in after Lloris had saved from Adam Lallana and Roberto Firmino.
At the Emirates Stadium, Sanchez brought the under-fire Wenger some breathing space as Arsenal overcame third-bottom Hull.

Arsenal's Alexis Sanchez scores their second goal from the penalty spot against Hull City
Reuters The Chilean forward contentiously scored with his hand in the 34th minute and netted a stoppage-time penalty after Sam Clucas had been sent off for handball to take his tally for the campaign to 17 goals.
It got Arsenal back to winning ways following successive defeats by Watford and Chelsea and left them in third place, nine points behind Chelsea.
Wenger dismissed a claim by former Arsenal striker Ian Wright that he had admitted his time at the club was coming to an end.
United closed to within a point of the Champions League places by beating Watford, Anthony Martial marking his return to the starting XI by scoring and setting up the opener for Juan Mata.
Jose Mourinho's side have now gone 16 league games unbeaten and while they remain sixth, they are just a point below both Liverpool and City.

Manchester United's Anthony Martial has his shot deflected by Watford's Sebastian Prodl
Reuters "Martial deserves the chance," Mourinho told the BBC. "He has worked harder than ever in the last two weeks in the way I like: silence, but hard."
He added: "In the first half the football was fabulous. When I got the job, they were saying at Manchester United you need to play football in a certain way and I agree."
Seventh-place Everton squandered a chance to close on the teams above them after being held to a 0-0 draw at sixth-bottom Middlesbrough.
Sunderland remain bottom of the table after Southampton's new Italian striker Manolo Gabbiadini scored a slick brace in a 4-0 win at the Stadium of Light.

Southampton's Manolo Gabbiadini celebrates scoring their second goal against Sunderland
ReutersJason Denayer also put through his own goal before Shane Long completed the scoring in added time.
Second-bottom Crystal Palace were also beaten, Joe Allen scoring in the second half to give Stoke City a 1-0 win that saw Mark Hughes's team climb to ninth.
Gareth McAuley was credited with a stoppage-time equaliser for West Bromwich Albion in a 2-2 draw at West Ham United.
After Nacer Chadli put West Brom ahead in the fifth minute, West Ham hit back through Sofiane Feghouli and Manuel Lanzini, only for McAuley to get a decisive touch to Jonny Evans's header at the death.
RESULTS:Liverpool 2 Mane 16,18
Tottenham Hotspur 0West Ham United 2 Feghouli 63, Lanzini 86
West Bromwich Albion 2 Chadli 6, McAuley 90+3
Manchester United 2 Mata 32, Martial 60
Watford 0Middlesbrough 0 Everton 0Stoke City 1 Allen 67
Crystal Palace 0Sunderland 0 Southampton 4 Gabbiadini 30,45, Denayer 88(og), Long 90+2
Arsenal 2 Sanchez 34,90+3(pen)
Hull City 0Standings P W D L F A
Pts1
Chelsea 24 19 2 3 51 17
592
Tottenham Hotspur 25 14 8 3 46 18
503
Arsenal 25 15 5 5 54 28
504
Liverpool 25 14 7 4 54 30
495
Manchester City 24 15 4 5 49 29
496
Manchester United 25 13 9 3 38 21
487
Everton 25 11 8 6 40 27
418
West Bromwich Albion 25 10 7 8 34 31
379
Stoke City 25 8 8 9 30 36
3210
West Ham United 25 9 5 11 34 43
3211
Southampton 25 8 6 11 28 31
3012
Watford 25 8 6 11 29 42
3013
Burnley 24 9 2 13 26 35
2914
Bournemouth 24 7 5 12 35 47
2615
Middlesbrough 25 4 10 11 19 27
2216
Leicester City 24 5 6 13 24 41
2117
Swansea City 24 6 3 15 29 54
2118
Hull City 25 5 5 15 22 49
2019
Crystal Palace 25 5 4 16 32 46
1920
Sunderland 25 5 4 16 24 46
19