He bowled in tandem with fellow spinner Jack Leach, a member of the England squad for the tour to Pakistan, as the pair got through more than 85 overs.
Foakes, who lost his England Test place to Surrey team-mate Jamie Smith, was caught at short leg for 37.
Vaughan then had Clark taken at mid-wicket by Tom Abell and Steel caught behind in the space of three deliveries.
Left-armer Leach did claim the vital wickets of Rishi Patel (70) and Al Hasan (12) before Tom Curran counter-attacked reaching 50 off 33 balls.
Leach ended the innings on the stroke of lunch when Curran was caught at long-on for 86 off 75 balls, including eight sixes, Leach finishing with 4-105.
Surrey began the match 24 points ahead of Somerset, with just two rounds of games left after this one, as they look to make it three County Championship titles in a row.