Asif was a late addition to the ODI squad at the behest of Test captain Misbah-ul-Haq, who felt the offspinner could come in handy against England's slew of left-handers, and proved his worth by taking 5 for 25 to bowl Zimbabwe out for 161 in 38.5 overs.
Zimbabwe had made an impressive start after losing the toss and being asked to bat, with openers Chamu Chibhabha and Richmond Mutumbami putting on 89 for the first wicket at a steady pace.
It was the 30-year-old's second cap for Pakistan after making a disappointing debut during Saturday's shock defeat in the second ODI, which Zimbabwe won by five runs on Duckworth-Lewis method.
Fellow spinner Imad Wasim also inflicted damage as he took 3 for 36 on his return to the side, and the only one wicket that did not fall to spin came with fast bowler Mohammad Irfan having Mutumbami caught behind for a career-best 67.