But hosts Exeter kept battling until the final whistle in search of a losing bonus point.
After a very fast opening half in the Devon sunshine at Sandy Park, Saracens led 10-6 after Owen Farrell broke through for the only try, which Charlie Hodgson converted.
Farrell then took over the kicking duties and, kicking into a light breeze, landed a penalty while fly-half Ignacio Mieres was on target with two of his three attempts for the Chiefs.
The former England and British & Irish Lions half-back scored what looked like being the only points of the second half as he converted his own touchdown.
But with time up on the clock the Chiefs took a quick tap penalty through replacement scrum-half Kevin Barrett and number eight Ritchie Baxter went over the line, with Mieres adding the conversion for the bonus point.
With just over 10 minutes remaining the Chiefs' openside flanker James Scaysbrook went off injured and, with replacement prop Alan Paver also off, they continued with seven players in the pack and uncontested scrums.
Exeter's head coach Rob Baxter had planned to make just one change to the starting line-up, with Aly Muldowney making his first start in the second row in place of fellow summer signing Peter Short.
But the Chiefs boss was forced to make one other change when New Zealand Maori prop Hoani Tui failed a late fitness test on a torn calf. Ireland A prop John Andress was promoted from the bench and Alan Paver, who is on loan from Cornish Pirates, was added to the replacements.
The visitors had South Africa international Schalk Brits back at hooker after he missed last weekend's victory over Leicester Tigers. James Short started on the left wing and Jamie George and Joe Maddock reverted to the replacements bench.
Saracens were on the scoreboard after just four minutes, with the referee playing the advantage as outside centre Farrell burst through for the touchdown, Hodgson adding the extras.
Mieres was just short and wide with a penalty attempt from just inside his own half a couple of minutes later but the Argentine made no mistake with his next attempt from 38 metres.
Exeter's outside centre Sireli Naqelevuki made a couple of breaks into the Saracens half, with the second taken on by right wing Nic Sestaret who cut back infield but was stopped 10 metres from the line.
Saracens were quickly back into the Chiefs half and Farrell, who had taken over the kicking duties from a slightly dazed Hodgson, landed a 35-metre penalty.
Going in to the final 10 minutes of the half, Saracens skipper Steve Borthwick was warned by referee JP Doyle that his front row had stood up twice at a 22-metre scrum and Mieres landed his second penalty.
The Chiefs mounted another attack down the right touchline through inside centre Jason Shoemark, but full-back Luke Arscott was stopped five metres from the line by Short.
With 40 minutes of the half completed on the clock Saracens were awarded another penalty on the Chiefs 22 and close to the left touchline, but Farrell's attempt went wide of the posts as the visitors led 10-6 at the break.
Both sides were matched for the opening 15 minutes of the second period but neither troubled the scoreboard.
Approaching the midway point of the half, Saracens kicked a penalty to the right corner and when the ball was worked into midfield Shoemark came out of the line and Hodgson went through the gap before adding the extras.
Five minutes later the Chiefs took the same option but from the resultant maul Barrett was penalised for pulling off Borthwick's scrum cap and the visitors relieved the pressure.
Despite being a man down and with uncontested scrums, the Chiefs went in search of another score as replacement wing Bryan Rennie chipped ahead to collect the ball but lacked support against a strong defence.
Exeter were finally rewarded for their efforts as Barrett took a tap penalty and, when he reached for the line and was stopped short, Baxter took on the charge for the vital touchdown which Mieres converted.
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