The pace bowler managed to get plenty of reverse swing on a flat pitch and finished with figures of 5-50 on day three as Sri Lanka were restricted to 283-9 at stumps on Friday.
However, there are concerns over the fitness of the 31-year-old, who appeared to hurt his already injured hand when fielding towards the end of the day and was in visible discomfort.
The majority of Sri Lanka's woes came in the middle order and, despite stubborn resistance from Upul Tharanga (83) and captain Angelo Mathews (89), the hosts closed 172 runs behind with just one wicket in hand.
Sri Lanka resumed on 30-0 but added just nine more runs before Steyn made the first break of the day when he tempted Kaushal Silva (8) to hook a bouncer to Vernon Philander at long leg.
Tharanga and Kumar Sangakkara steadied the ship with a stand of 59, but the tourists made a double breakthrough before lunch as Morkel bowled Sangakkara (24) and Steyn had Mahela Jayawardene - who recently announced he will retire from Test cricket later this year - for just three.
South Africa took just one wicket in the second session, but it proved a crucial one as Tharanga - who hit 14 fours and a six in his fine knock - was bamboozled by JP Duminy's flighted delivery and was stumped by wicketkeeper Quinton de Kock.
Skipper Mathews kept the scoreboard ticking over, but Steyn tore through his partners as he took the wickets of Lahiru Thirimanne (38), Dinesh Chandimal (6) and Dilruwan Perera (0), as Sri Lanka slumped from 190-4 to 201-7.
It looked as though Sri Lanka may make it to day four with Mathews still at the crease, but his steadying knock was curtailed when spinner Imran Tahir sneaked one under his bat and clattered the stumps.
And the situation worsened for the hosts with the final delivery of the day when Morne Morkel accounted for the wicket of Suranga Lakmal (6).