Morkel trapped the recalled Dawson lbw for a duck two balls later before England counter-attacked through Ali and Broad, who added 46 runs for the eighth wicket - mostly through swipes to the boundary from Broad.
Lallana is now due to report back to Liverpool for preseason training after being given a few extra weeks of holiday due to his involvement in England's recent (football) worldwide matches against France and Scotland.
Moeen and Stuart Broad - who already had the same double in safe keeping - were both in the runs and wickets to help England retain the upper hand.
But despite the absence of injured fast bowler Dale Steyn, South Africa - 2-0 winners in England five years ago - boast an impressive pace attack.
Neverthless, he ignored the pain and hit seven fours in his 52 before he was clean bowled by Moeen Ali attempting to deposit the off-spinner into the Pavilion.
England could have taken a stronger hold of the match but Quinton de Kock and Vernon Philander scored half-centuries to bring down the deficit below 100 runs.
By tea, Cook had struck six fours in his 50-ball innings, with South Africa wasting two reviews on England's former captain.
Alastair Cook made sure England do not throw away the position of advantage and the opening batsman actually strengthen his team's position in the opening Test against South Africa at Lord's.
Little went right for the tourists, in fact, until Rabada bowled Moeen through the gate driving and then won the fifth lbw verdict of the innings with one that kept low to pin Mark Wood on the back foot two balls later.
James Anderson was the last man out for 12, edging Morkel behind after sharing a 45-run last-wicket partnership.
The 90th over also saw him hit three fours off Broad - the best a whipped drive through mid-on from a ball pitching outside off stump - before his exit ended a stand of 66 in 75 balls with Philander.
Trailing by 458 runs, South Africa debutant Heino Kuhn didn't last long as he was sent packing by Broad for just one run.
England reached 119/1 at stumps to extend their lead over South Africa to 216 runs.
Broad then grabbed his second victim after nailing JP Duminy (15) lbw but Temba Bavuma and Theunis de Bruyn started to frustrate England as the duo knuckled down in the evening session.
Having batted for most of the first day, Root did not last long second time around, drawn into pushing at a delivery of ideal length from Morkel, which seamed a fraction to take the outside edge.