Goals from John Lundstram, Scott Wright and Ianis Hagi were complemented by a double from Kemar Roofe as the Scottish champions booked their place in the quarter-finals.
After losing their three previous games, including both legs of the Champions League qualifier against Malmo, McAllister felt the Rangers players gave manager Steven Gerrard exactly the response he was looking for.
“You are looking for a clean sheet, you are looking to score a goal and you are looking to perform,” said McAllister.
“I think we ticked all those boxes. We were looking for a reaction, a response and we got it. We started the game ever so well. The one word the manager was using prior to the game was ‘snap’. I thought there was a snap in our tackles, I thought there was a snap in our play and I thought we looked nice and sharp in front of goal. It has been a good night.”
Roofe’s contribution was warmly received by the Ibrox crowd as the English striker made his first appearance since experiencing the stress of spending five days in the Royal Hospital for Children in Glasgow with his baby son Cassius.
Happily the child is now back at home recovering and McAllister was full of praise for Roofe’s performance.
“I think things are better on that score (with his son),” said McAllister. “In the past week or so in training, we have seen the real Kemar Roofe. He is a player that has got over his little niggles, he is looking sharp in front of goal during the week in training.
“For me, when you train well you tend to bring it on to the pitch on a matchday and he is reaping the rewards of training really hard.”
McAllister felt Rangers’ overall performance was far closer to the standards they set last season in winning the Premiership title in such commanding fashion.
“We played with some nice confidence, there was some great combination play around the corner of the boxes to create the chances and we looked exciting,” he added.