Sri Lanka's World Cup match against Bangladesh was abandoned without a ball being bowled due to rain in Bristol on Tuesday.
Archer finished 3-52, his fifth three-wicket haul in six games, and pulled even with Australian paceman Mitchell Starc at the top of the tournament's bowling list with 15.
The 30-year-old, dropped following Pakistan's opener against the West Indies, made up for lost time as he demolished South Africa's beleaguered bowlers.
Hitting nine fours and three sixes in his 59-ball blitz, Sohail gave Pakistan a shot of adrenaline after Babar Azam's more sedate 69 laid the foundations.
Bangladesh remain fifth in the table, with the top four sides qualifying for next month's semi-finals.
Paceman Tim Southee missed out due to a calf injury as the Kiwis -- runners-up in the 2015 edition -- went in with two all-rounders in their starting line-up.
Henry returned figures of 3-29 and combined with fellow paceman Lockie Ferguson who also took three wickets to bowl out Sri Lanka for 136 in 29.2 overs in Cardiff.
The five-time champions bowled Afghanistan out for 207 in 38.2 overs on a warm day in Bristol and then made steady progress, reaching their target with more than 15 overs to spare, with Warner 89 not out.
Pakistan kept alive their chances of reaching the World Cup semi-finals on Wednesday, cruising to a six-wicket win against previously unbeaten New Zealand at Edgbaston.