Hill, who had to wait until the last minute before left-wing Kameli Ratuvou's try secured victory, said: "It's been a very special 15 years but it isn't about me, it's about the team."
The 34-year-old paid tribute to the medical personnel whose skill enabled him to return from the career-threatening injury which kept him out of the game for 18 months, left him with a permanent limp and effectively ended his hopes of defending the World Cup in 2007.
"There are many coaches throughout my career I would like to thank and I've also got a big medical list to go through," said Hill.
He added: "I've had a lot of highs away from Saracens in terms of playing for England and the Lions but then there have been the lows of injury in recent years.
"But there are no regrets. There have been setbacks but they have probably made me more determined to rise even higher."
Saracens director of rugby Alan Gaffney, who will be replaced by former Australia coach Eddie Jones next season, was far from happy with his side's performance in his last game in charge.
He said: "I thought we were dire. We started off well but seemed to go into snooze mode for 20 minutes.
"We just couldn't build any pressure. We've shown we can turn in good performances this season but we've lacked consistency and that's something we need to put right."
John Brain, Bristol's assistant coach, said: "The players created a winning position for themselves but didn't execute things correctly and take the right options.
"There were times when we needed to hang onto the ball in contact to put pressure on them but we didn't maintain that.
"We lost that game - Saracens didn't win it. Some people will say we were unlucky not to win the game but I don't buy that."
But Brain was pleased with the two-try contribution from 19-year-old centre prospect Luke Eves, saying: "He played very well and is a really exciting prospect for next season."