Electing to bat on a slow pitch at the Zahur Ahmed Chowdhury Stadium here, Bangladesh lost wickets just when the batsmen appeared to have settled down and was restricted to 240 for seven in their 50 overs.
Nazmul Hossain hit 73 off 82 balls to boost the total after useful contributions from Pinak Ghosh (43) and Joyraz Sheik (46).
The target proved too tough for South Africa, which was bowled out for 197 in the penultimate over despite a defiant 100 from opener Liam Smith.
Mehidy, leading his nation in the second successive ICC U19 Cricket World Cup, picked up three for 37 with his off-spin. Seamer Mohammad Saifuddin took three for 30.
It was Bangladesh's first win over South Africa in the ICC U19 World Cup following four previous defeats, but today's victory was its 12th in 15 matches against the same opponents over the past two years.
With Scotland and Namibia the other sides in the group, Bangaldesh's victory makes it the favourite to secure top place from Group A and avoid a possible quarter-final clash against Rahul Dravid-coached India.
Meanwhile, in a Group C fixture at the MA Aziz Stadium here, Dan Lawrence and Jack Burnham hit centuries as title contender England crushed Fiji by 299 runs.
England piled up 371 for three in its 50 overs against Fiji, with Essex's Lawrence smashing 174 off 150 balls with 25 boundaries and a six, and Burnham of Durham making 148 off 137 balls with 19 fours and four sixes.
The pair put on 303 for the second wicket after opener Max Holden had been dismissed in the third over.
Peni Vuniwaqa was the lone batsman to offer any resistance with 36 as four batsmen failed to score and five others did not reach double figures.
Lancashire seamer Saqib Mahmood grabbed three wickets for two runs in his five overs and his new-ball partner Sam Curran took three for 22.
England next plays the West Indies on Friday while Fiji takes on Zimbabwe the same day.
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