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.
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 Shanthakumaran Sreesanth also grabbed two wickets.
The three-match Afro-Asia Cup starts here on Wednesday, with the last two day-night games to be held in Chennai on Saturday and Sunday.
Africa XI 109 for 8 in 20 overs (Loots Bosman 52, Gulam Bodi 12; Mashrafe Mortaza 2 for 17, Sreesanth 2 for 26) lost to Asia XI 110 for 4 in 15.5 overs (Tillakaratne Dilshan 47 not out, Tamim Iqbal 30, Shoaib Malik 14 not out; Nehemiah Odhiambo 2 for 21) by six wickets.