The Potters eased to a 3-0 second round win over League Two Portsmouth, courtesy of brace of goals from Jon Walters and a late Peter Crouch header.
City will now face Sunderland at the Stadium of Light in the next round but Bowen was able to reflect on a stress-free night for Mark Hughes' men.
He told stokecityfc.com: "These games can be tricky as was proved last night when four Premier League teams went out of the competition.
"If you don't approach them right it can all go wrong for you on the night. In the build-up to the game it was all about the mentality of the group and making sure that whoever was selected to play was ready to go out there and make sure they did things professionally and did things right.
"I certainly think the result reflected that we got our approach right, as did the performance.
"I'm pleased for Jon that he got himself a couple of goals and I'm delighted for Peter that he managed to finish it all off with a goal right at the end because he was on the receiving end of a little bit of friendly banter from the Portsmouth fans.
"It's good for those guys to come in and do well and it's a reminder to the manager that they are ready to do a job when they get an opportunity."
Bowen reserved special praise for new loan signing Victor Moses, who marked his debut display with a 'Man of the Match' performance, and scholar Olly Shenton, who made his debut as a late substitute at the tender age of 16.
"Victor was a revelation, running at players and creating chances and creating a good goal for Jon," Bowen added. "For Olly to come on and get a taste of life on the first team stage was not only fantastic for him but also for the fans to see a local lad coming through.
"He's 16 but far more mature than that - he's got an old head on his shoulders and we've got real high hopes for him."
The Potters now face a long trip to Wearside to face Gus Poyet's Sunderland in round three. Bowen revealed: "We were looking for a home tie and were looking, if possible, to avoid a Premier League side. That hasn't happened so we've just got to get on with it."