Stuart Broad took four wickets for 12 runs before the tourists declared on 258 for nine, leaving England chasing a highly unlikely 259 from 39 overs.
Yet they took a calculated gamble by sending out Kevin Pietersen, whose 149 in the first innings changed the complexion of the game, to open.
He fell in the fourth over of the chase, hooking Vernon Philander to mid-on, and while Matt Prior was also sent up the order at No 5, once he was run out after a mix-up with Jonathan Trott England effectively had to give up their pursuit.
Trott and Ian Bell remained unbeaten until the close with the teams shaking hands on 130 for four after 33 overs.
"We felt it was worth having a go at it," Strauss told Sky Sports.
"We sent in KP, who is obviously in outstanding form, to set things in action but once Matt Prior got out it was probably time to shut up shop.
"It was a bit of a tall chase in the first place but it was certainly worth having a go."
Pietersen got England off to a fine start, crunching Morne Morkel's first over for three fours but once he was given out, it was Strauss who came out next.
"We just felt that if we let KP have a go, and then consolidate if he got out early, and then look at it a bit more and then try to get it into a Twenty20 situation towards the back end," Strauss said
"We probably lost a couple too many and once Matt Prior got out, it was too much to do."
Strauss can be pleased his side responded to their innings and 12 run drubbing in the series opener at the Oval in a determined manner although they still need to level the series in the decider at Lord's to cling on to their place atop the ICC Test rankings.
But the Middlesex man paid tribute to his side's character, particularly the bowlers, who had come under heavy fire for taking just two wickets in the opening Test against the Proteas. Broad's haul of five for 69 would have given South Africa something to think about.
"I was delighted with Stuart Broad today," Strauss said.
"It's nice to get into the South African order a little bit more.
"Our bowlers actually grafted really well for not a lot of reward.
"The ball went past the edge a lot but maybe those early inroads on the first morning is the key for us to win again."
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