Pakistan slumped to 291-7 at lunch, and after the break Agha Salman (20 not out) and Abrar Ahmed (17) tried to hit out for an unlikely win to keep the series alive.
But James Anderson had Ahmed caught, Wood dismissed Zahid Mahmood without scoring and Ollie Robinson ended the match with the wicket of number eleven Mohammad Ali for zero, sparking celebrations in the England camp.
Anderson and Robinson took two wickets apiece.
Shakeel and Nawaz's wickets proved the most consequential for England.
Shakeel, who hit eight boundaries in his 314-minute vigil, added 80 runs with Nawaz as England found wickets hard to come by on a Multan pitch that slowed considerably as the game progressed.
In a last-ditch effort, skipper Ben Stokes brought in Wood before lunch and was instantly rewarded with Nawaz caught by wicketkeeper Ollie Pope.
In his next over Wood forced Shakeel to play a pull off a short ball that caught his glove, with Pope taking a diving catch to his left.
Resuming on 198-4, Pakistan had lost Faheem Ashraf in the sixth over of the day for 10, caught at slip for spinner Joe Root's 50th wicket.