Off-spinner Bashir claimed 6-81 at Trent Bridge, his best figures in Test cricket, to dismiss the tourists for 255 in their second innings.
Early in the match, Zak Crawley (124), Ben Duckett (140) and Ollie Pope (171) all made centuries as the host piled up 565-6 declared.
It proved more than enough, with victory wrapped on the third evening and the perfect start to a big Test summer with India to come up next, which includes the second Test at Edgbaston. Last tickets now released. England Captain Ben Stokes: "It was good to get the summer off to winning ways. We were a little slow with the ball initially to get back into the game but then we were great from the last session of yesterday onwards.
"I felt really really good today, I felt in a good rhythm. I felt like it was coming out at a good pace which is something I wanted to work on.
"Another good week for me to get through, it gives me more confidence from what I already had."