Hampshire 135 (Clarke 5-29) trail Surrey 368 (Burns 151, Foakes 90, Edwards 5-93) by 233 runs
Burns, the Surrey opener, became the first man to reach 500 Championship runs this season on the way to making 151 against Steyn's Hampshire side. That left Hampshire following-on and Steyn, who is playing for them in a bid to prove his fitness ahead of the South Africa tour of Sri Lanka, deeply impressed.
"Rory Burns batted bloody nicely," Steyn told ESPNcricinfo. "He looked really good. I think he looks like a Test batsman. And I've opened the bowling to a lot of opening batters.
"When I blinked he was on 150, so he did it pretty quickly too. I always watch guys on the way they leave the ball and I thought he left the ball really well.
"It's always easy to come at the ball: you can nick one for four or hit it through the covers; it still says four in the scorebook. But I watch the way guys leave the ball and he left the ball with good intent. He watched it very carefully and, when the short ball came, he got out of the way or took it on."
Burns' century - his second of the season - provided the platform for a Surrey innings that saw only one other man make 50. And while he had a little fortune on the second day - he was badly missed at slip by Sean Ervine of Kyle Abbott when he had 126 - it was an innings that underlined Burns' patience, temperament and technique. Ben Foakes, with a stylish 90, was the only other man to offer a sense of permanence.
While this match was billed as a contest between the South African seamers - Morne Morkel is playing for Surrey and Kyle Abbott and Steyn are playing for Hampshire - it was actually a veteran English allrounder that stole the show.
Rikki Clarke, surely bowling as well aged 36 as he ever has, claimed the fifth five-wicket haul of his career and, in the process, earned Surrey an excellent chance of securing their third victory of the season. Hampshire, conceding a deficit of 233 on first innings, had just been asked to follow-on when bad light forced an early close.
But well though Clarke bowled (and the delivery that dismissed James Vince - pitching on off stump, lifting sharply and leaving the batsman - would have dismissed most), the key to this performance has been the relentless nature of the Surrey attack. But for an exploratory over from the spinner, Amar Virdi, the four Surrey seamers rotated throughout the Hampshire innings, each of them banging out an impeccable, probing length.
That has been one of the key differences between these dies. While Fidel Edwards bowled some lovely deliveries - and finished with a five-wicket haul - Hampshire were unable to sustain the pressure with their three main seamers going for at least 3.5 an over.
By contrast, none of the Surrey bowlers went for as much as three-an-over and, in the case of Clarke, many of the runs he conceded were thick edges to third man. Only one man in the Hampshire side - Rilee Rossouw - reached 15.
The days have has long since passed when Clarke might have won consideration by the England selectors. He would be the first to admit it took him some time to complement his talent with the rigour required to optimise it.
But, by luring him back to The Oval mid-way through last year, Surrey have gained the services of a highly experienced, hugely talented all-rounder who provides depth with bat and ball and can still catch the wind in the slips. It is 16 years since he last played a part in a Surrey side that won the County Championship; it may well be that he experiences it once more this year though those docile home wickets may yet come back to haunt them.
It won't stop them strengthening, though. Despite having Matt Dunn and Stuart Meaker in reserve - both are currently recovering from blows to the ribs sustained while batting - Surrey are one of at least six clubs understood to have made formal approaches to Leicestershire about talking to Zak Chappell. Warwickshire and Nottinghamshire are understood to be other front runners, with Nottinghamshire the favourites at this stage.
Such issues can wait. With two days left of this game, Surrey - through their relentless seam attack and Burns' high-class batting - have given themselves an excellent chance of victory. Hampshire, by contrast, have only Worcestershire below them and are up to their neck in a relegation battle.