The result at Eastlands narrowed the gap at the top of the table to just two points ahead of the Red Devils' clash with Middlesbrough.
On putting his managerial counterpart Sir Alex Ferguson under pressure, Grant said: "For the past three months we have won almost 80% of our league games.
"That is the record of champions and we have come a long way when you consider a lot of clubs were ahead of us during the season. But United are also doing well. I do not think, though, they will win until the end of the season. We will just carry on doing our job, like we did today.
Chelsea got off to perfect start and took the lead after only six minutes.
The home side bossed the rest of the first half but failed to make their possession pay.
He latched on to a through ball from Essien and rounded Hart to find the corner with his effort for his 11th goal of the season. Anelka almost added a third in the 55th minute only for his header to come back off the bar following a cross from Frank Lampard. It was just the response Grant was looking for following the Champions League first leg defeat by Fenerbahce in midweek.
Grant said: "You cannot play some players in three games especially when you come back from Turkey. Your body is not a machine and some players need more rest than others. I am pleased with the result today. I am always happy when we win.
"We started well but Manchester City came back at us. They have good team and a good manager.
"City put on a good show in the first half but we played better in the second period and could have scored more."