Bingham had led the Welsh qualifier 10-6 overnight and won the opening three frames of the final session to wrap up a convincing win.
Breaks of 117, 68, and 102 sealed an impressive victory and set up a last-eight tie against Anthony McGill.
The win marks Binghams first last-eight appearances since winning the event in 2015.
"I feel good and I am cueing well"
Following his victory, he said: "It was nice to get the job done quickly. I'm happy with my game. I could easily be 5-3 down and so to be 4-4 after the first session set me up.
The tables are playing really well and break-building is my game. I feel good and I am cueing well. But I need to sharpen up my safety game.
You have to be on your game here as you are playing top players every match."
Trump overcomes Gilbert
Trump had led 11-5 overnight only to see his lead narrowed to 12-8 after a fightback from Gilbert. Trump, however, won a scrappy final frame to wrap up the victory.
He said this after his win " It was a strange game. Dave was in control at the start and I was outplayed a little bit. I made a few mistakes.
He missed a great chance at 3-1 and after that the game kind of changed. I got some momentum and got a lead and after that, I was scoring pretty heavily.
I didn't play brilliant stuff but it was steady and hopefully, I can improve on that in the next round. I never really got any momentum or into any sort of rhythm. It was hard work"
Round of 16 results
Judd Trump 13 - 8 David Gilbert
Stuart Bingham 13 - 6 Jamie Jones