Grant said: "He can be as good a goalscorer as Drogba. He's done a good job until now throughout his career and he can be even better with a big club.
"I think Anelka can take the chances that will make the difference in matches.
"He knows how to score goals. He has good pace and, in the modern game, you need three or four strikers in the team."
Chelsea have paid £12m up front and will fork out another £3m to Bolton based on Anelka's appearances over the next three years.
The striker, 28, has agreed a three-and-a-half-year deal worth around £90,000-per-week. This will be his eighth club in 11 years but Grant is confident he will finally make his home at Chelsea.
The Blues boss said: "A few weeks ago I decided to go for Anelka. He's a player who wants to finish his career at a good club and win some titles, which is important to us.
"You can look at the positive or the negative. He's been at so many clubs but, at 28, he's a good age for a footballer and says he wants to finish his career at Chelsea."