Neither team looked capable of making the breakthrough in a game dominated by defences, but that all changed when Birch came off the bench with 18 minutes to go.
The striker, without a goal in two months, had only been on the pitch for 60 seconds when he grabbed the winner.
An excellent run from youngster Lee Frecklington set-up Oliver Ryan who saw his shot blocked by the legs of Scott Shearer. The ball spun up in the air and Birch was first to react heading home from six yards.
Until that goal neither keeper had a save to make, although Rovers created the better of the chances.
They were unlucky in the 31st minute when Richard Walker got clear but saw his shot blocked by Colin Cryan.
Then a cross from Lewis Haldane split the Lincoln defence but neither Walker or his strike partner Junior Agogo could finish.
Rovers had a great chance in the 68th minute when a mistake by Lincoln skipper Jamie McCombe let in Agogo.
The striker found himself in the clear on the edge of the box with Lincoln keeper Alan Marriott stranded of his line but the Rovers man sent his shot wide of an empty net.