The Asian team were struggling at 34-3 before surpassing their opponents' modest total of 109-8 with more than four overs to spare in the day-night game on a seamer-friendly pitch.
Bangladeshi teenager Tamim Iqbal smashed a 22-ball 30 before middle-order batsman Dilshan steered his team home with a 49-run stand for the fifth wicket with skipper Shoaib Malik (14 not out).
Left-handed opener Iqbal, 18, hit one six and three fours, while Dilshan struck one six and six fours in his 36-ball knock.
Kenyan seamer Nehemiah Odhiambo had jolted the Asian side with two wickets in his lively opening spell.
Opener Loots Bosman top-scored with 52 for Africa XI, who had been put in to bat on a track with pace and bounce.
The South African struck three sixes and as many fours in his 55-ball knock.
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South African Gulam Bodi (12) was the other batsman to reach double-figures.
Bangladeshi seamer Mashrafe Mortaza was the most impressive bowler with 2-17 off four overs. Indian paceman Shanthakum