Fifth-placed Manchester City play Bournemouth on Monday (Tuesday AEDT) when victory would leapfrog them into second.
At the other end of the table, the bottom three all lost with 20th-placed Sunderland thrashed 4-0 at home by Southampton, for whom new striker Manolo Gabbiadini scored twice, and 19th- placed Crystal Palace going down 1-0 at Stoke City.
Liverpool were without a league win in 2017 and had been knocked out of both domestic cups during a recent slump, but rediscovered their verve to outplay a lacklustre Tottenham whose nine-game uneaten league run came to a halt.
Senegalese Mane, whose absence on Africa Cup of Nations duty coincided with Liverpool's slump, scored in the 16th and 18th minutes.
Liverpool boss Juergen Klopp was delighted as his side moved into fourth place, a point behind the two north London clubs and ahead of Manchester City on goal difference and 10 points behind Chelsea.
"Today the pressure was on us but we dealt with it," Klopp said.
"We had a really bad period. We are still around, not with Chelsea but with the rest."
Alexis Sanchez scored both goals for Arsenal at the Emirates, although his first in the 34th minute owed much to luck as the ball went into the via his hand.
"The referee apologised to us after halftime and said it was a handball," Hull defender Andrew Robertson said.
The Chilean's second also came via a goalline handball, this time from Hull's Sam Clucas in stoppage time, which led to a penalty and sending-off.
Sanchez's spot-kick took his tally to 17 goals in 25 league games, including four against Hull.
At Old Trafford, Manchester United extended their unbeaten league run to 16 games.
Juan Mata put them ahead before Anthony Martial doubled their lead after the break. Fifteenth-placed Middlesbrough earned a 0-0 draw to move two points clear of third-bottom Hull while mid-table West Bromwich Albion had Gareth McAuley's last-gasp equaliser to thank for a 2-2 draw at West Ham.
