However, Milan failed to build on Ronaldinho's third-minute opener and only Clarence Seedorf's flicked goal five minutes from time ensured the tie remains up for grabs going into the second leg on 10 March.
After a lull in the tempo of the match, Hodgson introduced Dempsey from the bench in place of Stephen Kelly and with practically his first touch - a header from a Davies cross - he would have put Fulham ahead overall for the first time in the tie but for Chimenti's save.
Branston is ineligible for Northampton's FA Cup replay with Darlington next Tuesday as he was not signed in time for the original tie, but he does have permission to play in the second round if Gray's men progress.