Drogba nipped in to intercept Gerrard's back pass to put Chelsea ahead and Frank Lampard added a second after the break.
Drogba told Sky Sports 1: "He made a big mistake. I was lucky to be there and score. It was good because this goal gave us a lot of confidence to play well."
The Ivory Coast striker claimed referee Alan Wiley should have given the visitors a penalty when Salomon Kalou appeared to be fouled.
Drogba added: "It's not easy to make a decision like this but when it is a penalty you have to give it."
Man-of-the-match Lampard was pleased to get a difficult match out of the way.
He said: "This was a huge game for us, if we had dropped points today we would have given the advantage to Man U."
Lampard also had some sympathy for Liverpool, who were knocked out of the Europa League at the semi-final stage by Atletico Madrid on Thursday.
"It's been a difficult week for them, getting knocked out of the cup, and they gave everything they had."
Chelsea coach Carlo Ancelotti said his side deserved their victory.
"It was a very important step for our fight for the title," he added. "We played very well, we deserved to win.
"We are happy because it was a difficult step against Liverpool. Tactically we did very well."