Jim Mallinder hailed his players for silencing the Ashton Gate crowd as Saints ruined Bristol’s first home game back in the big time.
Tries from George North, Louis Picamoles and Ken Pisi put Saints in control at 25-3 up at half-time.
And after Ross McMillan had replied for Bristol after the break, Mike Haywood put the finishing touches on the Saints win, scoring the bonus-point try four minutes from the end.
It was a convincing display from Saints, who were far sharper than they had been in the previous week’s 18-14 defeat to Bath at Franklin’s Gardens.
And Mallinder said: “Overall we’re pleased to get five points and get in winning ways after last week’s defeat.
“We needed to win today and our first half was very good. Our set piece went well, we put the ball in the right areas, put them under pressure and made them make a lot of mistakes.
“We were disappointed not to get a bonus point just before half-time with Dylan Hartley not scoring that try that could have killed the game off, but credit to Bristol - they came back into it and they showed their spirit they’ve got.
“In the second half, we didn’t dominate field position as much as we should have done but ultimately we got there in the end and we’re pleased to get the five points.
“We needed to come here and shut the crowd up and put massive doubt into Bristol’s mind that they could play at this level. That was our intention and I think we did that quite well.”
Meanwhile, Mallinder says Dylan Hartley will be fit to face Saracens on Saturday after being withdrawn at half-time at Bristol.
Hartley came off with a sore back at the break, but Mallinder insists taking the England captain off was purely precautionary.
He said: “Dylan’s got a little bit of a sore back and we’ve got Mike Haywood back after not playing last week and we thought it was a good time to swap that over.
“Dylan’s had a bit of a sore back for a couple of weeks so he’s just been looking after himself.”
George Pisi was forced off early in the second half, with Harry Mallinder coming on in his place.
And Mallinder said: “George Pisi felt his hamstring and he had a slight niggle there, but hopefully it’s not too bad.”