Having won the final four frames on Friday for an 11-5 lead, Trump moved within a frame of victory after a 67 and sealed the match with a run of 69.
In the last eight he will play Graeme Dott, the 2006 champion and last year's runner-up, who beat Ali Carter 13-11.
Mark Allen will face Mark Williams after a 13-12 win over Barry Hawkins.
Allen won six frames in succession, compiling runs of 79, 114 and 82 in three consecutive frames to open a useful advantage going into the evening session.Trump capitalises on Friday night as Gould misses final black
The Northern Irishman then opened a 12-9 lead but was pegged back to 12-12 before clinching the decider with a break of 96 to sweep into the last eight.
Trump, who won the China Open this month and defeated reigning world champion Neil Robertson in the first round, again looked in assured form.
The Bristolian admitted his recent success has been difficult to comprehend but believes he can go on to win the world title on 2 May.
"I always knew if I brought a decent game with me I could go out and win it, and I still feel that," he said. "I'm definitely super-confident. I feel like I've got the game to go out and win it.
"I'm on such a high. I've got used to winning whereas I was used to losing before.
"The last month has exceeded my wildest expectations. I have to bring myself back to reality sometimes. I've only won two games, so it's nothing that special. I've still got a long way to go. There's a lot of tough players out there to beat."
Gould agreed that the world number 11 had the capabilities to lift the trophy and said: "If he scores like he has been and keeps potting the long ones like he does, then who knows, his name could be on the title.
"We've all known Judd for many a year, and we all know what he can do."Behind the scenes at the Crucible with Michaela Tabb
Dott and Carter had been locked at 8-8 overnight and the Scot appeared on course for a 147 in the 20th frame but lost position after the 10th red and the break ended on 93.
The match looked set to go to a deciding frame with Carter holding a useful 52-21 lead in frame 24, but Dott was allowed back in and maintained his composure on a tricky final black to seal a 38 clearance, punching the air in jubilation.
Carter, runner-up at the Crucible three years ago, said: "I lost five or six frames that I should never have lost in a million years.
"I was absolutely robbed in a couple of them. That last frame was just a joke really. I'm very disappointed. I gift-wrapped him 5-3 in the first session and you can't do that with top players.
"All credit to Graeme, he's played well. It was a good match, it's just a shame I ended up on the wrong end of it. I've had a few matches like that in my career. That's got to be looked at I think."
Dott offered a similar verdict, saying: "In the first session I played terribly, and how I managed to win it 5-3 was unbelievable.
"Ali will be gutted. He's got to be. I think he cued better than me. I stole so many frames. I must have won about five on the black with clearances, but you don't practise with John Higgins over all these years and not learn how to do it. He's a machine at doing that."Allen wins final frame in thriller
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