The Owls were gifted the chance to take the lead inside two minutes. Cameron McGeehan's misplaced pass found its way to Kadeem Harris whose initial shot was blocked only to fall for debutant Jacob Murphy who calmly slotted the ball home for his and Wednesday's dream start.
Steven Fletcher wrapped it up after an hour when Kieran Lee's square pass fell to him and he dropped a shoulder to create space before striking a fine effort into the bottom corner from the edge of the area.
Bullen said: "We set out a game plan and we stuck to it right to the very last seconds of the game. We watched Barnsley last week against Fulham and they pressed Fulham to death.
"So, we thought our game plan was to do the same today. I felt the players did that to a tee. The easy option when you get your second goal is to sit back and just soak up the pressure.
"But we put our substitutes on who had a magnificent effect on the game as well and just gave us the energy we needed. If you're being greedy, we could have scored one or two more with the opportunities we had.
"We've got 11 players who were excellent but if we are going to focus on the two wide men, they gave them a lot of trouble all day.
"It might be old fashioned that I play wingers who want the ball and stay wide but they're out-and-out wingers, so we want them to do the natural thing. I want to keep the game simple."
Barnsley manager Daniel Stendel said: "I'm more upset with the performance. We need to congratulate Sheffield Wednesday for the deserved victory. We didn't play as well as we can play.
"It wasn't the best start from us, conceding in the first minute in our first away game but Wednesday played a simple but effective game.
"Normally we can play so much quicker when we win the ball, but we didn't have a chance today to use our counter-attack and we lacked conviction.
"I think it was more about the lack of experience in this game. We haven't got many players with experience in the Championship.
"We signed players with no experience in the league and no experience in English football. This is the risk we took but it's only one defeat.
"We have so much more to show but it's a big lesson. They want to learn and sometimes it's hard when you lose but it's the best experience to learn."