The home side trailed 10-9 at the interval but two tries early in the second half gave the hosts the impetus they needed to close out the bonus-point win.
But Leicester's home win over Bath meant the result was not enough to secure a home semi-final in the Aviva Premiership play-offs for Sarries, who now face a tough trip to the Tigers next weekend for a place in the final.
Exeter, who had already secured their place in next season's Heineken Cup prior to today, were first on the scoreboard when Gareth Steenson gave them a third-minute lead with a 35-metre penalty.
The opening exchanges were littered with errors, but on 12 minutes Owen Farrell levelled the scores with a 45 metre penalty.
A careless knock from James Short in his own 22 gave the Chiefs an excellent opportunity to score the first try but Sireli Naqelevuki was bundled into touch just two metres short.
The hosts knocked on again at that lineout though, and from the resulting scrum Tom Johnson forced his way in under the posts for Steenson to convert and make it 10-3.
Farrell then missed a glaring chance to reduce the arrears when he failed to convert a 30m penalty attempt from straight in front of the posts.
But he was on target with two penalties in quick succession to close the gap, taking his side into the break 10-9 down after Steenson missed with a 45m attempt before half time.
Saracens made a dream start to the second period, scoring twice in seven minutes, the first coming when Bryan Rennie's chip ahead fell straight into the hands of Short, who raced in unopposed from 80m.
Sarries then capitalised when the Chiefs lost a lineout in their own 22 and Rhys Gill charged through some weak Exeter tackling to score. Both tries were converted by Farrell.
Farrell's fourth penalty extended Saracens' lead to 26-10 going into the final quarter before Farrell created the try for Goode and then converted.
Charlie Hodgson then produced the moment of the match when he waltzed through a gap in the visitors' defence before rounding replacement full-back Ignacio Mieres, for a brilliant score.
Farrell converted to bring his match tally to 20 points but Exeter launched a rally with two late tries from Chris Budgen and Matt Jess with one conversion from Steenson, but it was too little too late.
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