Shakib struck his sixth ODI century and took 4-41 with the ball to help Bangladesh to an 87-run victory in Chittagong, giving the hosts a 1-0 lead in the five-match series.
Shakib and Mushfiqur shared a 148-run fifth-wicket partnership that lifted their team out of trouble, and the Bangladesh captain was quick to praise his partner, who hit 101 off 99 deliveries.
"Credit goes to Shakib," Mushfiqur said. "I took my time and Shakib was going from ball one so that helped me.
"We knew if we got to 250, it was par and our bowlers bowled really well as well.
"One of us had to have stayed till the end and that was why we have won.
"The boys are really hungry. We want to win the next match as well, all of us are focused on that."
For Zimbabwe captain Elton Chigumbura, there was disappointment after a promising position had been thrown away.
Having taken four early wickets, Zimbabwe were unable to stem the flow of runs from Shakib and Mushfiqur.
And with the bat in reply, they looked well set at 146-4, but then lost their final six wickets for just 48 runs.
"I thought the start we had was excellent but when Shakib and Musfiqur came in, they played really well," Chigumbura said. "Our fielding was off as well, we didn't take our chances.
"One guy was scoring a hundred, another was hitting a fifty for them and that's where we need to improve.
"Some of the balls were spinning, but it was a good decision to bowl and it was a chaseable score but we kept losing wickets.
"Obviously, we have four more games and we need to bounce back, especially the next one."