Patel scored a fluent 42 on a turning wicket with six boundaries before England was bowled out for 306 in its first innings with offspinner Shoaib Malik having career-best figures of 4-33.
Resuming at 222-4, England lost both its overnight batsmen — James Taylor and Jonny Bairstow — in quick succession before Patel put on 40 runs with Rashid Adil.
Malik had Rashid caught close to the wicket just before lunch and then clean bowled both James Anderson and injured Ben Stokes after the break to fold the England innings.
Yasir Shah, who knocked Patel's top of the off stump with a sharp leg break, claimed 3-99.
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