Marco Jansen fell just short of a maiden Test fifty before Anrich Nortje added a useful 28 not out as South Africa were dismissed for 326 in reply to England's first innings 165 -- a lead of 161 runs -- on the third day of the first Test at Lord's on Friday.
Only one side has won a Test at Lord's after conceding a bigger first innings deficit, with England coming from 171 runs behind to defeat South Africa by 71 runs back in 1955.
South Africa led by 124 runs overnight after a rapid stand of 72 in just 75 balls between Jansen (41 not out) and Keshav Maharaj (41) had followed a painstaking 73 by opener Sarel Erwee.
But they lost Kagiso Rabada without adding to their total off Friday's third ball of the day when his pull off Matthew Potts was brilliantly caught one-handed by a leaping Broad at mid-on.
Stokes, who had rocked South Africa with a burst of two wickets for five runs in seven balls on Thursday, took up the attack alongside Durham team-mate Potts.
The new-ball was taken as soon as possible, with South Africa 291-8 off 80 overs but Stokes did not immediately bring senior pacemen James Anderson and Broad -- England's all-time leading Test wicket-takers -- into the attack.
Broad heeded a spectator's pleading shout of "pitch it up!" after coming on six overs into the new ball when he had Jansen edging a drive to Zak Crawley in the slips, the towering fast bowler out for a 48 that included four fours and a six.
England, however, will have to bat considerably better than they managed in their first innings if they are to retain a realistic chance of going 1-0 up in this three-match series.
They have won all four of their Tests under a new leadership duo of Stokes and coach Brendon McCullum, but each of those wins have come as a result of batting aggressively in the fourth innings rather than having to set a total for their opponents to chase.