The late wicket of Jacques Kallis kept England in the hunt after three days of the final Test, with South Africa leading by 139 at Lord's.
South Africa 145/3 and 309 lead England (315) by 139
Kallis and Hashim Amla, who remained unbeaten on 57 at the close, had threatened to take the game away from England with a partnership of 81.
But Finn trapped Kallis LBW in somewhat controversial circumstances - the batsman convinced he had got an inside edge - to renew England's hope of winning the match and holding onto their no.1 Test status.
Jonny Bairstow fell five short of a first Test century earlier in the day, but England's tail wagged to hand them a slender six-run lead - Graeme Swann making an unbeaten 37.
And while Hashim Amla, nightwatchman Dale Steyn and a strong South Africa middle-order stand in their way, the pitch has shown little sign of deterioration and England will harbour hopes of chasing even a stern target.
Ebb and flow
England negotiated their first target of the day with some ease, Bairstow and Matt Prior easily playing the eight overs remaining of the old ball.
Prior though threw all that good work away the moment Vernon Philander took the new cherry, chucking the kitchen sink at a loosener outside off stump and edging it straight to Kallis at second slip.
Bairstow made his way pretty serenely into the 90s, showing the kind of temperament which had so impressed on the previous day. But Morne Morkel has been excellent in this Test and he bowled a wonderfully tight spell, drying the runs up with the result that Bairstow played across the line to a straight one and had his stumps rattled.
It was reward for a fine spell and when Steyn removed Broad it looked as if South Africa would take a decent lead.
But Graeme Swann is nothing if not determined and, alongside first James Anderson (who took a painful blow to the finger) and then Steven Finn England scratched their way to a slender six-run lead. The sight of Finn pulling Steyn to the square-leg fence produced one of the biggest cheers of the day.
The prospect of seeing the tourists 54/4 for a second time in the match never seemed likely, and the manner in which Graeme Smith and Alviro Petersen started would have made England's hearts sink.
But Graeme Swann settled into a nice rhythm, particularly with the left handed Smith on strike, and after Prior had wasted a review for a caught behind appeal, the off-spinner got his man - Smith plumb LBW playing a sweep. Broad had a rare bowl from the Pavilion End and added Petersen moments later, trapping the opener LBW with one which nipped back down the Lord's slope. Amla was also lucky to survive after gloving one down the leg side, but Prior couldn't quite hold onto what would have been a great catch.
Kallis, like Tendulkar, Lara and Ponting before him, belongs to the club of greats to have struggled at the Home of Cricket, and it seems likely to remain that way. For the second time in the match he was left disgruntled with the DRS, convinced he had got an inside edge on a good ball from Finn.
But there appeared to be no evidence of willow on leather for the third umpire to reverse the decision, and the great all-rounder departed in furious fashion - though the Snickometer later suggested a thin inside edge.