The English had Soumya Sarkar (zero) out in the third over after winning the toss and electing to bowl in Townsville. But a 122-run partnership between Litton Das (102) and Anamul Haque (56) was to follow as Bangladesh set themselves up for a grandstand finish.
Off-spinner Sam Wood (1-39 off 10 overs) broke the partnership though, before the re-introduction of left-armer Reece Topley (5-32 off 10) saw Bangladesh collapse in the final stages. Topley removed Salman Hossain (26), Al-Amin (three), Mosaddek Hossain (nine) and Nurul Hasan (zero) as Bangaldesh finished at 217-7 from their 50 overs. Wood (46) and Ben Foakes (47) batted well for England, adding 84 runs for the second wicket in just 85 deliveries.
Al-Amin (1-25 off eight) removed Foakes at a crucial period though as England lost four wickets for 48 runs in just under 15 overs. But captain Adam Ball (34 not out) and Aneesh Kapil (33 not out) combined for a 60-run stand for the seventh wicket that saw England home as they finished at 218-6 with 10 balls to spare.
In the Plate Championship, a superb Javed Ahmadi (134) century saw Afghanistan set a huge total of 336 for seven from its 50 overs against Scotland. Ahmadi scored his runs from 111 deliveries, hitting 17 fours and four sixes, while Najibullah Zadran (83 off 51) also swung hard to give them a score that Scotland never looked likely to chase down.
Mathew Cross (37) and Peter Ross (34) were Scotland's best but they never got near Afghanistan's total as Aftab Alam (three for 63 off 9.4) helped them to a 126-run victory, with the losers bowled out for 210 in the 40th over.
Papua New Guinea won a thriller against Zimbabwe by 12 runs, also in Brisbane, with the winners batting first and posting a score of 235 all out from their 50 overs.
Charles Amini (63) and Lega Siaka (50) helped Papua New Guinea to that score, while Luke Jongwe (three for 38 off seven) and Kieran Geyle (three 44 off 10) bowled best for Zimbabwe.
A 108-run partnership inside the first 20 overs from Luke Masasire (68) and Geyle (42) looked to have set up victory for Zimbabwe but they did not count on Kabua Vagi Morea (3-26 off 10) and Christopher Kent (2-26 off seven) bowling so well.
And Zimbabwe lost its last five wickets for just 32 runs as they were bowled out for 223 with just five balls remaining.
The day's other match in Brisbane saw Nepal score 219 for seven with the bat against Namibia, with Pradeep Airee (98 not out) playing a starring role.
And left-arm spinner Bhuvan Karki (5-21 off 7.3) starred as Namibia were bowled out for 180 in the 47th over.