The home side had a good chance to take the lead when Dylan Carreiro's corner was not cleared by the East Stirlingshire defence and the ball fell to Bobby Linn, who could only fire his strike wide of the target.
Arbroath continued to threaten as Paul McManus fired a shot at goal from distance which was only parried by Richard Barnard. Simon Murray tried to latch on to the rebound, but he put his strike wide from a tight angle.
In the second half, the hosts threatened again this time through Adam Hunter, but his effort from distance was well saved by Barnard.
Hunter caused the visitors problems again minutes later as he raided to the edge of the box, but his strike this time curled just past the post.
East Stirlingshire took the lead in the 72nd minute when Gilmour collected the ball and fired home from close range, beating the outstretched hand of Marc McCallum.
The home side pushed to get an equaliser late in the game, but couldn't find a way past Barnard as they were condemned to their seventh defeat of the season.