The fast bowler struck twice as England lost seven wickets wickets for 43 runs in 93 balls to be 182 for eight in their second innings at lunch.
Jonny Bairstow, dropped on seven, was 28 not out and Mark Wood unbeaten on nought.
One consolation for England, who now had a lead of 279 runs, was that history was in their favour.
Only three sides have made more than 200 to win in the fourth innings of a Lord's Test.
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Former England captain Alastair Cook was 59 not out overnight and Gary Ballance unbeaten on 34.
The pair found runs hard to come by against a South Africa attack now back to full strength with Vernon Philander returning after being unable bowl Saturday having been hit on the hand while batting against James Anderson.
Cook, in his first Test since resigning the captaincy had added just 10 runs to his score when he drove uppishly at Morkel and was well caught in the covers by Temba Bavuma.
Ballance hung his bat out and was caught behind for 34 as the towering Morkel took two wickets for one run in 10 balls.
Root, fresh from his first innings 190, fell for just five when, having seen Keshav Maharaj turn one sharply, he played on to a straighter ball from the left-arm spinner.
Ben Stokes (one) was then so plumb lbw to fast bowler Kagiso Rabada he 'walked' before the umpire's finger was raised.
The one disappointment for South Africa in the session was when Bairstow lofted Maharaj to long-off, where Philander dropped a seemingly simple catch under grey skies.
The ball also crossed the boundary rope -- one of four fours in five balls that Bairstow struck off Maharaj.
Dawson bagged a pair when bowled by a Rabada full toss and Broad was caught at short leg off Maharaj.