Having drawn 0-0 with the Puritans in the first qualifying round of the Buildbase FA Trophy at Hayden Road on Saturday, Diamonds battled back from a goal down to win the replay thanks to second-half goals from Tom Lorraine and Jordan Macleod.
It has set Peaks’ team up for a long trip to BetVictor Northern Premier League leaders South Shields in the next round a week on Saturday.
And the Diamonds boss was delighted with the way his team “finished the job”.
“I was really pleased because we finished the job we started on Saturday,” Peaks said.
“I would class Banbury as a good team and in the three encounters we have had with them over the past week, we have more than held our own.
“They were better than us in the first half but I thought our tactical changes in the second half really paid off.
“We should have won at the weekend so to go there and do the job is really pleasing. It’s as good a win as we have had in recent times.”
Diamonds, who return to Southern League Premier Central duty at Biggleswade Town this weekend, now face a mammoth trek to the north-east in the next stage of the Trophy on Saturday week.
And while Peaks is relishing the tough challenge in front of his team, he believes it is “ridiculous” that such a long journey is having to be made at this stage of the competition.
“I would say it’s probably the hardest draw we could get at this stage,” he added.
“And I think it’s a bit ridiculous that we are having to travel that far at this point in the competition.
“But it should be a great day and weekend and we will make the most of it.
“The Northern League sides are always hard to play against but we will go there and try to perform to our best and if we do that, then we will give ourselves a good chance.”