Andre Villas-Boas's team were on course for a valuable three points when Belgian defender Jan Vertonghen struck twice for the visitors either side of halftime after Luis Suarez gave the Reds a 21st minute lead.
But Stewart Downing capitalised on woeful defending from Spurs right-back Kyle Walker to equalise in the 66th minute.
And England star Gerrard netted the winner from the penalty spot in the 82nd minute after Benoit Assou-Ekott fouled Suarez.
Tottenham remain third, but fourth-placed Chelsea, two points behind, and fifth placed Arsenal both now have a game in hand in the race to finish in the top four.
Brendan Rodgers' Liverpool climb to sixth place, closing the gap on Chelsea to seven points, and retain an outside chance of qualifying for the Champions League.
In the day's other Premier League match, Senegal striker Papiss Demba Cisse bagged a stoppage-time winner as Newcastle boosted their survival bid with a 2-1 victory over Stoke at St James' Park.
Alan Pardew's team moved nine points clear of the Premier League relegation zone thanks to Cisse, who struck two minutes into additional time to complete a fine fightback from the Magpies.
Jonathan Walters put Stoke ahead from the penalty spot with 67 minutes gone, but Yohan Cabaye's superb free-kick soon after gave Newcastle renewed hope.