The Sarries fly-half kicked five penalties and a drop-goal.
Boss Eddie Jones said: “It was scratchy but a pretty good effort after three games in nine days.
“There are players who are absolutely exhausted, physically out on their feet.”
Bristol were only 11-10 down at half-time but when fly-half Adrian Jarvis’ radar went off line with three misses the hosts were sunk.
Coach Richard Hill said: “At times we ran Sarries to a standstill.”