Mohammed Nabi plays a shot during the T20 match at the Sher-e-Bangla National Stadium on Sunday. AFP
Nabi hit an unbeaten 84 off 54 balls under pressure to guide Afghanistan to 164-6 before Mujeeb claimed a career best 4-15 to help his side dismiss Bangladesh for 139 in 19.5 overs for their world record 12 successive Twenty20 international win.
Mujeeb took three wickets in his opening spell to reduce the home side to 32 for four before Mahmudullah and Sabbir Rahman put a little resistance with 58 runs for the fifth wicket.
Gulbadin Naib broke the partnership by removing Mahmudullah, who made 44 off 39 balls for Bangladesh before Mujeeb ended Sabbir's innings at 24 in his final over to surpass his previous best of 3-17 against Ireland.
Naib claimed 2-27 while skipper Rashid Khan and left-arm pacer Fareed Malik also took two wickets each.
Nabi earlier helped Afghanistan recover from 40-4 with his fourth Twenty20 international fifty, which featured three fours and seven sixes.
He received great support from former skipper Asghar Afghan as the duo shared 79 runs in the fifth wicket stand.
Asghar made 40 off 37 balls after he was reprieved on 31 when Sabbir took a catch at the deep only for the reply to show that bowler Taijul Islam had overstepped.
Nabi and Asghar overshadowed the bowling heroics of Mohammad Saifuddin, who claimed a career best 4-33.
"It's a team effort. Bangladesh bowled brilliantly in the powerplay and that put the pressure on us. Me and Asghar planned well and in the last five overs the plan worked," said man of the match Nabi.
Bangladesh skipper Shakib Al Hasan also did well with the ball to claim two wickets for just 18 runs.
"I thought we were brilliant in the first 10 overs. They scored more than 100 runs in the last 10 overs, that cost us the game. We didn't play well with the bat to chase those runs," Shakib said after the match.
Both Bangladesh and Afghanistan won their respective opening match of the tournament against Zimbabwe.
Bangladesh: Shakib Al Hasan (capt), Liton Das, Soumya Sarkar, Mushfiqur Rahim, Mahmudullah Riyad, Afif Hossain, Mosaddek Hossain, Sabbir Rahman, Taijul Islam, Mohammad Saifuddin, Mustafizur Rahman.
Afghanistan: Rahmanullah Gurbaz, Hazratullah Zazai, Asghar Afghan, Najibullah Zadran, Najeeb Tarakai, Mohammad Nabi, Rashid Khan (capt), Gulbadin Naib, Fareed Malik, Karim Janat, Mujeeb Ur Rahman.
Umpires: Masudur Rahman (BAN), Gazi Sohel (BAN)
TV Umpire: Sharfuddoula (BAN),
Match Referee: Chris Broad (ENG)
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