Resuming the fourth day at 70 without loss, Stoneman was bowled for 59 and Burns, who scored a century in the first innings, followed suit for 66.
Ollie Pope and wicketkeeper-batsman Ben Foakes took the total past 200 before the experience of Morne Morkel and Rikki Clarke ensured they recorded a ninth consecutive Championship victory this season.
South Africa fast bowler Morkel has been an influential figure with the ball for Surrey and it was his five wickets for 24 that limited Worcestershire's second innings score to 203 on the third day.
Surrey, who have now won 10 and drawn two of their 12 matches, have an opportunity to complete an unbeaten season, with a trip to Somerset and a home match against Essex to come.
(Reporting by Hardik Vyas in Bengaluru; Editing by Keith Weir)
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