The most yet made by any side to win in the fourth innings of a Test at Old Trafford is England's 294 for four against New Zealand in 2008.
South Africa's already difficult task was not helped when, after rain briefly delayed the start of their chase, they slumped to 18 for two.
Dean Elgar, who made a gritty second-innings hundred in England's 239-run victory in the third Test at The Oval, fell for five on Monday when caught behind off Stuart Broad.
Fellow opener Heino Kuhn (11), struggling to cement his Test place, repeatedly played and missed before edging James Anderson, again bowling from the newly-named James Anderson End, low to Alastair Cook at first slip.
The experienced Amla battled away, taking 40 balls to score his first boundary when he whipped off-spinner Moeen Ali for four.
But on the stroke of lunch, Bavuma was caught behind off Toby Roland-Jones, although England had to review umpire Kumar Dharmasena's original not out decision to confirm a thin nick.
Earlier, England were bowled out for 243 in their second innings after resuming on 224 for eight.
Ali, dropped on 15 by slip Elgar -- arguably the easiest of five catches the Proteas put down on Sunday -- top-scored with 75 not out after being unbeaten on 67 overnight.
Morne Morkel wrapped up the innings by dismissing tailenders Broad and Anderson during a spell of two wickets for one run in four balls.
The tall fast bowler finished with fine figures of four for 41 in 13.1 overs.
But Ali's 66-ball innings, including nine fours and three sixes, maintained England's grip on this match.
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