The former Arsenal striker joined the Gunners' north London rivals last month, after an unhappy spell at Manchester City, and netted away at Wolves to open his account.
Two more at White Hart Lane confirmed his threat to opposition defences, and any fears the Tottenham faithful would not take to a former Arsenal favourite have been dispelled.
"I'm very happy, very delighted," Adebayor told Sky Sports.
"It's always important to score your first goal at home.
"I have done it already away.
"I know the fans all love me because that's what they're looking for, a player who can score goals.
"I'm very happy that everything is going well for me at this club.
"Hopefully I can keep up the hard work and keep going."
Red cards for Charlie Adam and Martin Skrtel left Liverpool depleted and Adebayor's two goals came with the visitors down to nine men.
"What is important is to take three points," Adebayor said.
"We did that well with lot of qualities. I'm delighted for the team."