All-rounder Ben Stokes took the final two wickets in the first 25 minutes of play to secure the man of the match award in what was an engrossing contest from first ball until last.
But for his presence in the England side Bangladesh would have been celebrating that historic victory in Chittagong.
The first test between England and Bangladesh remains finely poised with Bangladesh requiring 33 runs with two wickets remaining to take the lead in the two-match series.
Taijul Islam (16) notched the first boundary of the morning three balls into the second over, gloving an attempted hook off Stokes over Jonny Bairstow to the delight of the few raucous home fans in the stadium.
Ben Stokes got England a much-needed breakthrough when Taijul Islam was trapped in front of the wicket!
Two balls later and Shafiul Islam fell the same way, this time given out on the pitch and confirmed by replays when he requested a review, as man of the match Stokes had the final say on proceedings.
England are keen to reiterate no-one is being dropped - Broad bowled an especially impressive spell towards the end of the first Test, including nine consecutive overs at the end of the fourth day - but may feel that Rashid, who was not at his best in Chittagong, would benefit from another bowl ahead of the India series.
Since the retirement of Andrew Flintoff almost a decade ago, England have been lacking a top-quality pace bowler who can also bat up the order and give them an ideal balance to the side.
Stokes steadied England's second innings after the top order collapsed on day three and Vaughan said England can not afford to put themselves in a similar situation against the world's top-ranked Test side. "And just finding a method of making sure I use that as well as I possibly could, whether it be changing my angles on the crease or where I deliver the ball from". Credit to both sides.
"I was fairly confident this morning, if I'm brutally honest", he said. I genuinely thought that 280 was going to be enough.
This is another fine example of why you need five days in a Test match. Bayliss believes Stokes's inclusion in the side helps the team play with three pacers and three spinners, a luxury for touring sides in the subcontinent.
With star paceman Jimmy Anderson now sidelined, Stokes is emerging as one of the keys to England's chances on a marathon tour of the sub-continent which will see them play five Tests in India in November and December after wrapping up a two-Test tour of Bangladesh.
Mushfiqur said: "Nobody expected Bangladesh to play in this manner after 15 months so we had to prove that we could do something".
Durham all-rounder Stokes has hit form at the right time, with England moving on to play India in five Test matches once their tour of Bangladesh ends at the conclusion of the second Test in Dhaka, live on Sky Sports.