Hill, who made nine changes from the side that lost at Saracens and welcomed scrum-half Shaun Perry back from England duty, watched his side end a run of four straight league defeats with tries from Rob Higgitt, Bernardo Stortoni, David Hilton and Roy Winters.
Ben Cohen, Sean Lamont and Martias Cortese scored Saints' tries.
"If I had not rested some of our old guard for last weekend's game at Saracens I don't think they would have had the legs to cope with the pace of Northampton," said Hill.
"We just about had enough energy in the tank to survive though we were beginning to tire.
"Whenever we play them Saints play a certain type of game and when they are behind they begin to open up which is when they are at their most dangerous and they cut sides to shreds. We should have buried them several times in the first half.
"If we had lost this game I think the momentum may have gone to see us through but I think that we need another win against Worcester and pick-up at least a bonus point from the other two against Leicester and Gloucester to take us through to the play-offs."
Saints head coach Paul Grayson, who joined the club as a player in 1993 and has tasted relegation in the past, is still unclear about his future at the club.
But the former England fly-half remains committed to the club and still hopes they can avoid the drop with three matches to play.
"We left some points out there for sure and we should have scored four tries and come away with a minimum two league points," said Grayson.
"We have no intention of giving up without a fight. There are no excuses for the position that we are in and we are certainly not proud of where we are but that is the reality of it.
"It is a pattern that has repeated at the club over the years but we have a month to sort things out and we are not down yet.
"We built-up the score for Bristol but we are still waiting to score the four tries. We did get across the whitewash three times and there are not many teams who will do that here."
Rookie fly-half Danny Gray, who kicked 13 points in the game, was drafted in for his first Premiership start of the season and Hill was pleased with the performance although the 23-year-old half back will not be with the club next season.
Hill has already completed his recruitment plans for next season, including centre Neil Brew and No.8 Andrew Blowers who are already at the club having arrived early.