Trump - who was 9-1 on to progress - won the next two frames, but he was clearly out of touch as the Welshman steeled himself with a closing 68 enough to see him complete the biggest win of his career.
Ronnie O'Sullivan had no such trouble as he made breaks of 120, 66, 90 and 99 on his way to a 4-2 victory over China's Yuan SiJun.
The 18-year-old won the first frame before the four-times Welsh Open champion turned on the style to book his place in the last 32.
Sijun was unfortunate in the sixth frame as he potted a yellow only to see a red drop as O'Sullivan punished him heavily to advance.
He will face Alexander Ursenbacher in his quest to reach the last 16.
World champion Mark Williams is out of the event after suffering a 4-2 defeat to Zhang Anda.
The two-times Welsh Open champion - winner of his home event in 1996 and 1999 - opened up with a break of 55, but was forced to sit and suffer as world number 73 Zhang made 71 on his way to a 3-1 lead.
Williams produced 83 to close to 3-2 behind, but Zhang killed it off in the penultimate frame.
World number seven Kyren Wilson was also an early casualty after he suffered a 4-3 defeat to world number 103 Joe O'Connor, produced runs of 88, 53, 66 and 65 in ousting the German Masters winner.
O'Sullivan moves onto 993 career centuries, but could easily be on 995 with two obvious century attempts being turned down by the Essex player, who made 90 in the fourth frame only to be penalised for a foul when potting a red as the referee tried to re-spot the pink.
Jones completes victory over Trump
Jack Lisowski won a deciding frame against Marco Fu with a sparkling century to reach the third round of the Welsh Open.
The 27-year-old raced into a 2-0 lead before Fu hit back to level the scores following a break of 87 in the fourth frame.
Lisowski kept his composure though, with a total clearance of 140 to go 3-2 up before Fu set up a decider by winning the next frame with a battling effort of 71.
‘This is the very best of Jack’ - Lisowski makes 140 against Marco Fu
Lisowski then topped things off with a superb break of 122 in the decider.
World number one Mark Selby eased past Adam Duffy 4-0 with breaks of 93, 128 and 67.
Elsewhere, former world champion Ken Doherty lost 4-1 to Sam Craigie while Craig Steadman beat Soheil Vahedi by the same scoreline.