A mismatch looked on the cards after fifties by captain Harry Brook (53) and Jacob Bethell (55) fired England to 184-7, which appeared beyond the modest batting resources of Nepal, a tier-two member of the International Cricket Council.
Rohit Paudel and his teammates, however, refused to throw in the towel and put up a spirited chase only to fall agonisingly short of a famous victory.
Kushal Bhurtel (29), Paudel (39) and Dipendra Singh Airee (44) kept them in the hunt before Lokesh Bam spread panic in the English camp with his late batting blitz.
England's spin trump card Adil Rashid was taken out of the attack after three expensive and wicketless overs while Bam hit speedster Jofra Archer for back-to-back sixes, showing scant regard for reputation.
Pressure mounted but Bam never took his foot off the pedal to put the pressure back on England and propel Nepal to a position where they needed 10 runs from the last over from Sam Curran.
Curran bowled with ice in his veins, conceding only four runs in five deliveries to set up a last-ball thriller with Bam at the striker's end looking to seal a famous win for his team with a six.
Curran fired a low full toss, denying any elevation to Bam, who could only hit it to the deep cover for a single.
"It wasn't easy at all but thankfully we got over the line," a visibly relieved England captain Brook said afterwards.
"I thought we were in a good position to defend that. I didn't think it would go as close as that. Hats off to the Nepalese team, they played extremely well."
Bam remained unbeaten on 39 as Nepal finished on 180-6.
After electing to bat at the Wankhede Stadium, England slumped to 57-3 in the seventh over to find themselves in a spot of bother.
Bethell had no problem adjusting to the nature of the track, where other batters found shot-making difficult.
Bethell and Brook combined in a 71-run partnership during their 45-ball collaboration to rebuild their innings.
Will Jacks provided the late surge for England with an unbeaten 39 off 18 balls, which proved crucial in the end.
The batter smashed four sixes, including three in the final over from Karan Chhetri.
Bhurtel led Nepal's charge when they returned to begin their chase and, after his departure, Airee and skipper Paudel kept Nepal in the hunt with a stand of 82 from 54 balls.
"Disappointment is there but at the same time, I'm very proud of the team," Paudel said.
"We had belief when we came in this World Cup. We didn't come here to participate, we wanted to compete and we had that belief in the group that we can beat any of the sides if we play our best brand of cricket.
"I thought today we played our best brand of cricket."