City finished the game strongly after David Silva's double and could have won more comfortably, but Platt felt the hosts were not at their best.
He said: "We scored after 90 seconds and you think that should settle us. But we didn't really get going at full throttle in the first half. In the second half we were better."
City hope the win will put title rivals United under some pressure ahead of their tricky game at Tottenham on Sunday.
Platt said: "We have got 15 (games) left. Both teams will drop points, I am sure.
"We have got to make sure we keep getting victories so if United slip up we are in a position to take advantage of that."