After trailing to a 21st-minute Jordan Mustoe opener at the break, Westley saw his side produce a much-improved second-half performance to score four times.
Substitute Akpo Sodje bagged a brace while Graham Cummins and Paul Huntington were also on target.
Westley said: "We want to do well in this competition and after going 1-0 down I thought the lads responded superbly well in the second half to earn the win.
"We made a few changes and gave some of the lads who have not been featuring a game and I thought they responded superbly and were a credit to themselves.
"I wasn't happy with their first goal and our defending at times is leaving a bit to be desired, but there were more positives than negatives and I was delighted with the appetite they showed to take us a step closer to that Wembley final which we would all love."