The co-hosts looked to be probably heading out of the competition with England progressing to the last eight when they slumped to 169-8 chasing 226 to win on an unpredictable wicket with the ball reverse swinging.
But Shafiul decided attack was the best way ahead and he blasted 24 in as many balls to see his team over the line for the unlikeliest of wins as the home crowd went wild.
Earlier, Jonathan Trott top-scored for England with 67 backed up by Eoin Morgan (63) to help their side recover from a poor start to reach 225 all out in the Group B clash.
The result was an amazing turnaround for the home side who were skittled for just 58 against West Indies a week ago in Dhaka before losing by nine wickets.
"You never know with cricket. It's unbelievable," said vice-captain Tamim Iqbal. "We knew that he (Shafiul) could bat but not like that." Mahmudullah (21) and Shafiul added 58 runs in their unbroken ninth wicket to take their team to victory with an over left.
Bangladesh had been apparently cruising toward victory thanks to solid innings from Imrul Kayes (60) and Tamim Iqbal (38).
But they suddenly lost momentum with the victory in sight as five wickets tumbled for 14 runs before Shafiul and Mahmudullah rescued their country.
Ajmal Shahzad was comfortably the best of a very mixed England attack, taking 3-43 from his 10 overs.
England now need to win their final match against the West Indies (March 17) to be sure of a quarter-final place while Bangladesh next face Netherlands on Monday and then South Africa on March 19. Wins would take them through.
England skipper Andrew Strauss looked stunned — and bemused — after losing their second Group B thriller against a second tier nation. They were also beaten by Ireland yet defeated South Africa.
"We wanted to win this game and win it well. We struggled a little bit with the bat. I thought 225 was a competitive total. We needed to bowl and field well but didn't do that as well as we'd have liked," he said.
"We got ourselves in a great position to win the game but in the end we couldn't take those two final wickets. We're bitterly disappointed.
