The Tigers did the damage in the first half as they kept up this season’s pleasing habit of scoring early, netting twice in the opening six minutes at Prenton Park with Jon Toral and David Milinkovic on the scoresheet, before Ryan Tafazolli added a third just before the break.
Rovers responded well in the second half but the Tigers showed their defensive strength to claim a first clean sheet of the season to progress.
Speaking after the game, Honeyman said: “It felt good. You can never quite prepare for a goal inside the opening 20 seconds but you take it every time.
“We played really well in the first half and to go into the break with a three-goal lead was fully deserved on the back of our performance in the first 45.
“It was always going to be difficult to maintain those standards and keep the same intensity in the second half when you’re three-nil up.
“We knew there would be a reaction from Tranmere and it was pleasing to see us grind it out and – importantly – keep a clean sheet.
“It was a strong defensive effort after the break and full credit to the lads at the back and Matty in goal – they all played their part.
“Overall, to score three goals away from home and keep a clean sheet – you’ve got to be delighted with that.
“It was pleasing to get the job done. We’re delighted with the win and to be in the next round.”
A trip to Championship rivals Preston North End is the Tigers’ reward for winning at Prenton Park, with Honeyman eyeing an extended run in the competition.
“They’re in our league and it’s going to be another tough game,” he added. “But, we want to go as far as we possibly can in the cup.
“The gaffer is very keen on getting a winning mentality at the Club and that means going out in every competition trying to win every game. We’ll look forward to the challenge and hopefully get into the third round.”