A crowd of 1161 packed into the Windmill with the expectation that Jason Cadden's side would get the win they needed but Chasetown who are looking for a play-off place had other ideas.
A goal from Perrow in the sixth minute had the visitors ahead and it stayed this way until the 64th minute when Stein netted. Six minutes later, Birch added a third.
Leamington did reply with a goal from Stanley in the final minute.
Second placed Nuneaton Town won 2-0 at Atherstone Town. In front of a bumper crowd of 1450, Boro kept up the pressure on Leamington with first half goals from Dean and former City youngster Dave Blenkinsopp.
Bedworth United beat Malvern Town 3-0 with goals from Hill, Kinder and Gordon.