Goals in either half from Rhian Brewster and Billy Sharp, who replaced the former Liverpool youngster off the bench, saw the Blades move up to 13th in the table and kick off the Heckingbottom era with three points.
Heckingbottom replaced Slavisa Jokanović late last week, with the Blades hierarchy signalling the club was taking a new approach after sacking the former Fulham and Watford manager.
And Heckingbottom said: "I was pleased with lots of things in the game. I thought the approach was good and the players responding to the messages was very good.
"I think it was important we got that goal before half-time. We were easily good value for one-nil - it should have been more.
"The second half was very similar - 21 shots and it's still one-nil and you're wondering if they're going to get a chance at the other end.
"When that second goal went in, it was an important moment in the game.
"We knew Bristol's threats to be on the counter with a long ball. In open play, we were defensively strong.
"Bristol came to slow the game down and the players' intensity, I thought, was good. We managed to try and keep the game played at our tempo when the ball was on the pitch. That was an important thing for us."