The Gunners - who have secured just one win from eight in all competitions - went ahead through Bacary Sagna on the hour mark.
The Blues are now within touching distance of the champions, who have to visit Stamford Bridge at the end of April.
Grant - who replaced Jose Mourinho in September - reflected: "I have always said to the players when we were chasing Arsenal and Manchester United, many points behind them, that we only wanted to be able to be in this position - that when they come to Stamford Bridge it can be a game which if we win we can go ahead of them.
"Today, we did that.
"We played good football from the beginning and we responded very well from going behind.
"I don't know if we can win the title now, but we will be doing our best to do it.
"It is important we play attacking football and score goals; for me, this is more important than any other way - and I will keep doing things in the way that I am."
Some sections of the Stamford Bridge faithful chanted 'you don't know what you're doing' and called the name of their former manager when Grant elected to make a double change with 20 minutes left.
Those same supporters were soon cheering as the substitutes played their part in both of Drogba's goals.
Grant, who gave captain John Terry instructions on how to reshape the team, said: "When you make a substitution, I do not think any manager believes it will succeed every time.
"We wanted to be able to play in two systems, and I think this was the time to change.
"We had chances but did not score, so we wanted to put more pressure on the defence.
"I did not hear the crowd. I am concentrating on what I need to do and am very confident in what I am going to do - because I know the team and how they are."