Bristol captain Fitz Harding scored three tries to put Bristol in the driving seat - scrum-half Harry Randall and wing Toby Fricker also touched down - but Saracens showed admirable resilience.
Elliot Daly scored two tries, with wing Rotimi Segun and flanker Toby Knight also scoring, while Lozowski kicked 17 points.
Saracens boss Mark McCall hailed an "amazing backs-to-the wall victory".
"I have been at the club a long time and I can't remember a win like that one," he said.
"There were so many occasions when it felt like they were on top, and somehow there was this fighting spirit among the group."
Fly-half AJ MacGinty landed two penalties and two conversions for Bristol.
His penalty restored an eight-point advantage for Bristol just before the hour mark but Saracens remained in the contest as an absorbing encounter moved towards its closing stages.
When Daly pounced for his second try with nine minutes left, again converted from wide out by Lozowski, the game was back in the balance.
And Lozowski then settled the contest in dramatic fashion.
Bristol rugby director Pat Lam said: "I am proud of the effort, the way we played. We blew a lot of opportunities and one thing with Saracens is if they stay in, they have a chance. That is their history.
"Probably at the end, we tried to close the game out too early when we had a scrum. It is a good learning. But I cannot fault the effort."
In the later game on Saturday, Francois van Wyk scored a late try to give Bath a 26-24 win over Harlequins, lifting them to second in the table.
Quins led until the 77th minute of a seven-try thriller at The Stoop but were never able to relax against last season's beaten finalists, who showed their resilience to stay in the fight, with Thomas du Toit crossing before Van Wyk struck in the left corner to snatch victory.
