The tourists, who lead the three-match series 1-0, reached 23 without loss in their second innings with Sarel Erwee on 12 and Dean Elgar on 11 but they face a tough task as the Old Trafford pitch favours seamers and spinners.
Stokes and wicketkeeper-batsman Foakes had put on a 173-run partnership for the sixth wicket after South African quick Anrich Nortje struck twice to remove England's overnight batsmen during an outstanding opening spell in the morning session.
Jonny Bairstow fell one run short of his half century, with Nortje adding reverse swing to his top pace and the Yorkshireman edging to Erwee at first slip.
Another fine delivery ended the solid contribution of 38 from under-scrutiny England opening batsman Zak Crawley, with Nortje's perfect line inducing an edge to keeper Kyle Verreynne.
That meant England were on 147-5, still four runs behind South Africa's first innings total. But Stokes and Foakes then batted positively before lunch to begin building the lead.
Stokes showed his intent by hitting spinner Simon Harmer over mid-wicket for six, while Keshav Maharaj had an lbw decision against Foakes go in his favour only for it to be overturned on review.
After lunch Stokes brought up his half-century by hitting Simon Harmer over mid-on onto the England dressing room balcony but he kept his aggression in check for most of his innings.
Stokes accelerated to his 12th test ton, reaching the landmark in 158 balls, before he found the leading edge off Kagiso Rabada and was caught by Elgar at cover.
There were cameos from Stuart Broad (21), Ollie Robinson (17) and Jack Leach (11) but Foakes ended unbeaten on 113 after his second career test century and his first on home soil.
South Africa lead the three test series 1-0 after their victory in the opening match at Lords.
