Costa, who has already found the net seven times since arriving from Atletico, has received praise for settling so quickly at Stamford Bridge, and Drogba is the latest person to reveal his admiration for the 25-year-old.
"Like me, he came to this level a little bit late. When I look at him, I can see the desire to win every challenge, to score goals. He's very good," the Ivorian told reporters. "He's hungry for goals, and that's the best thing for a team like us.
"I think for him to start the season in England, coming from Spain, which is a very different league, and to recreate the same thing he was doing at Atletico, is fantastic.
"That's the best feeling he could have. He's the striker we needed this season."
Chelsea will look to extend their lead at the top of the Premier League when they meet Manchester City on Sunday.