Both players hit their career-best knocks to frustrate Afghanistan on Wednesday, Afif posting an unbeaten 93 - his maiden half-century - while Mehidy was left 81 not out.
Their brilliance led Bangladesh to race to victory with seven balls to spare after Afghanistan had been bowled out for 215 in 49.1 overs.
"Honestly, I did not believe we could win after 6-45," Bangladesh captain Tamim Iqbal said.
"But I'm very proud and happy for the two for their incredible innings. Afghanistan have a brilliant attack but Mehidy and Afif did well."
Afghanistan's spin trio of Rashid Khan, Mujeeb Ur Rahman and Mohammad Nabi - described by Tamim Iqbal as world's best - could take only two wickets, with Afif and Mehidy keeping them at bay.
Playing only his second ODI, pacer Fazalhaq Farooqi had left Bangladesh reeling at 4-18 as Afghanistan appeared to have quickly closed out the game.
Mujeeb (1-32) took the wicket of Shakib (10), who was the only one of Bangladesh's top six to reach double digits.
Rashid Khan (1-30) then claimed the wicket of Mahmudullah with a half tracker, edging Afghanistan closer to their seventh straight ODI victory before Afif and Mehidy rose when it mattered most.
Najibullah Zadran had earlier struck a superb 67 off 84 balls to give Afghanistan's total respectability after they'd won the toss and decided to bat.
Mustafizur Rahman finished with 3-35 while Shakib Al Hasan, Taskin Ahmed and Shoriful Islam took two wickets each.
Bangladesh are hosting Afghanistan for three ODIs and two Twenty20 internationals, with the next ODI being on Friday.
