Firmino's effort in first-half stoppage time ended Liverpool's barren run of 482 minutes without a league goal.
Alexander-Arnold doubled their lead in the 47th minute before Pierre-Emile Hojbjerg quickly lashed in his first goal for Spurs to give the hosts hope.
With Tottenham's defence unravelling, Mane struck again for the champions in the 65th minute and there was no coming back for Jose Mourinho's side whose disappointment was compounded by an injury to talisman Harry Kane.
Both sides also had goals ruled out by VAR decisions.
Liverpool's seventh successive win over Spurs in all competitions moved them into fourth spot with 37 points from 20 games, four points behind leaders Manchester City who have played a game less. Tottenham stayed sixth with 33 points.
This was not quite Liverpool back to their best but it was an important statement of intent for Klopp's side that they will not be giving up their title without a fight.
MAKESHIFT DEFENCE
Even with a makeshift central defensive partnership of Jordan Henderson and Joel Matip they were too strong for a Tottenham side who were culpable for all three goals.
"Nothing has really changed, we're the same team we always have been," Alexander-Arnold said.
"We have the same mentality going into games, we respect the opposition. We have the expectation on ourselves that we should win the game and it was a whole performance from the lads."
Liverpool should have ended their drought in the opening minutes when Mane was through on goal but steered his shot wide.
That miss looked costly as Son Heung-min thought he had given Spurs the lead from Kane's pass but his effort was ruled marginally offside by VAR.
While Tottenham looked dangerous on counter-attacks, Liverpool were the more threatening side and they were gifted the opening goal deep into stoppage time.
Mane again got behind Tottenham's defence and prodded the ball across towards Firmino who capitalised on indecision by keeper Hugo Lloris and Eric Dier to tap home.
Jose Mourinho made a double change at half-time, with Kane and Serge Aurier replaced by Erik Lamela and Harry Winks, while Nathaniel Phillips took Matip's place at the heart of Jurgen Klopp's back four.
Alexander-Arnold doubled Liverpool's lead two minutes into the second period with a clinical right-foot finish following a mistake from Lloris, who palmed Mane's shot into his path.
Hojbjerg quickly got Spurs back into the match with a rasping long-range drive before referee Martin Atkinson disallowed a Mohamed Salah goal for handball against Firmino.
Liverpool were not to be denied a third goal after 65 minutes, though, the excellent Alexander-Arnold whipping in a cross that Joe Rodon failed to deal with and Sane punished the defender by lashing home with his left foot.
The Brazil international has been a key performer for Jurgen Klopp's side at the heart of defence this season, helping to fill the void left by the injured Virgil van Dijk.
Fabinho has helped Liverpool stay within touch in the Premier League title race but they went into the Spurs game seven points behind leaders Manchester City.
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