However Matt Tubbs struck for Salisbury in three minutes later and Rob Sinclair made it two just after the break.
Danny Crow got a consolation for the U's 10 minutes from time but Salisbury held on to progress.
City will face Tamworth at home in the next round on 20 February.
Cambridge boss Martin Ling told BBC Radio Cambridgeshire:
"There's times when I've made excuses for them but there's no excuse tonight, it's not good enough.
"Confidence is obviously low because results aren't going for us.
"It just seems there's a death wish out there and we seem to be able to lose a game at any given stage at the moment."