Afghanistan scored 145 for six after electing to bat, a target which they defended riding on Rashid's brilliance.
Bangladesh were well on course to winning the inconsequential match after senior pros Mushfiqur Rahim (46 off 37) and Mahmudullah (45 not out off 38) stitched 84 runs for the fifth wicket.
In fact, after the penultimate over of Karim Janat which yielded 21 runs, Bangladesh were the firm favourites to win the duel but Rashid single-handedly turned the game upside down by defending nine runs off the final over.
With Bangladesh needing 30 runs off the last two overs, Mushfiqur struck Janat for five consecutive boundaries to bring down the equation to nine off the last over.
But then Afghan captain Asghar Stanikzai went to his trump card Rashid, who responded brilliantly by removing a set Mushfiqur with his first delivery and then did enough with his guile to snatch the narrow one-run win for his side.
Besides Mushfiqur and Mahmudullah, no other Bangladesh batsmen could make any substantial contribution.
Even though he didn't take too many wickets, Rashid (1/24) eventually turned out to be the difference between the two sides.
Earlier with the series already in its bag, Afghanistan wasted a good start to be restricted to a modest 145 for six.
Electing to bat, Mohammad Shahzad (26) and Usman Ghani (19) made a solid start, sharing 55 runs off 46 balls for the opening stand before Bangladeshi bowlers struck twin blows in consecutive overs to put brakes on their opponents.
Shahzad, who hit three fours and one six, was the first to depart caught plumb in front of the wicket by left-arm spinner Nazmul Islam in the eighth over.
Ghani, who struggled a bit, followed suit in the next over, edging one to Mushfiqur behind the stumps off right-arm pacer Aby Jayed's bowling as the Afghans slumped to 59 for two in 8.5 overs.
Skipper Stanikzai played a short little cameo scoring 27 off 17 balls with the help of three sixes before perishing in the 13th over.
Samiullah Shenwari remained unbeaten 33 off 28 balls while Najibullah Zadran made 15 towards the end to take Afghanistan close to the 150-run mark.
Islam (2/18) and Jayed (2/27) were the successful bowlers for Bangladesh in the inconsequential match.
Prior to Thursday's match, Afghanistan has already taken an unassailable 2-0 lead in the series, having won the first two matches by 45 runs and six wickets respectively.
First Published: June 8, 2018, 7:50 AM IST