Arsenal defender Bacary Sagna had put his side ahead in the 59th minute when he headed a Cesc Fabregas corner past goalkeeper Carlo Cudicini from close range.
Drogba then capitalised on some poor defending from Kolo Toure to hook in a deep cross into the box after a knock-on from Nicolas Anelka - and win it with just eight minutes left.
And the Ivory Coast international, who was named man of the match, is confident his side can move forward after this result.
"I think they give the belief we can still be in the race against Manchester United," Drogba told Sky Sports 1.
"Arsenal are not dead even though they lost this game. Although it is going to be difficult for them and their confidence will be hit.
"We played well today, it is always difficult against arsenal. But we are strong and I think we deserved the win today."
Drogba also praised the impact of substitutes Nicolas Anelka, John Mikel Obi and Juliano Belletti, who came from the bench in the latter stages to give the Blues a final push.
"I think the substitutes did their job and that is what it is all about - the team. They wanted us to show that even though they are on the bench at the start they can be important.
"I hope we can win the title this year and the way we celebrated showed we believe that we can do it."
He added: "William gave some good years to Chelsea and the minimum the fans should show him is some respect."