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Owen Farrell could have handed Saracens the lead, but he also sent his kick from distance wide of the uprights.
Ulster went ahead when Paddy Jackson showed accuracy with his boot to kick the home side ahead in the 15th minute, while he then extended their advantage with a drop goal.
The boot of Jackson proved to be deadly for hosts again when he fired a penalty through the posts to hand the home side a nine-point cushion.
However, Chris Wyles continued his strong start to the campaign by bringing Saracens back into the match when he crashed over in the corner for the first try of the match, although Farrell sent his conversion wide of the posts.
Farrell missed another kick before the break to leave the visitors trailing by four, but Wyles and Chris Ashton almost combined at the start of the second half to hand Richard Wigglesworth a try-scoring chance, but the winger's grubber kick went inches out of the scrum-half's reach.
Billy Vunipola became the second Saracens man to enter the sin bin during the contest for colliding with a player, but the visitors were able to take the lead through good play from Duncan Taylor as he released Alex Goode to touch down on the line, while Farrell added the extras.
Taylor notched Saracens' third try of the match after he finished a flowing move following good work from Farrell to create space for the centre to find the line, but the fly-half missed the conversion.
Farrell added a penalty to all but secure the win, and the visitors scored their fourth try to bring up the bonus point as Vunipola bounded over the line following his return to the pitch to claim an emphatic victory.