Trump edged a series of tight frames through the middle part of a match which failed to deliver the fireworks many had hoped for, pulling into what appeared to be an unassailable 4-1 lead.
Robertson took the following two frames and had chances in the next, but a costly missed black allowed Trump a second bite of the cherry and this time he made no mistake.
It wasn't Trump as his fluent best by any means, but with the World Championship little more than a month away this may prove to be a significant victory against arguably the player of the season so far.
"I started OK to be honest," said Trump. "Neil was missing a couple and I was able to pick up the pieces. I had a good opportunity to win 5-1, and after that just felt a bit nervy to be honest. I just wanted to get the match over and done with.
"In the end I needed him to gift me it, really."
Robertson was emotional after the game, the Australian clearly knocked off his stride following the news that cricket legend Shane Warne had passed away.
"No," said Robertson when it was put to him that he had never got going. "(I was) hit by a truck, with the news leaving my hotel about Shane Warne, you know, a national hero and an idol of mine.
"As much as I tried today, I just couldn't do anything. The more I tried to delete what happened from my brain, the more it was in there. I couldn't concentrate, couldn't think properly, couldn't do anything. Completely deflated.
"I fought well, I tried hard. I've never felt like that before. It's just awful."
Hossein Vafaei awaits Trump in Saturday's semi-final after the Iranian beat Zhang Anda 5-4 in his last eight match.
Lisowski fightback stuns Carter
Jack Lisowski progressed to the semi-finals of the Welsh Open with a brilliant comeback that saw him beat Ali Carter 5-4.
Given his Ukrainian heritage, Lisowksi's mind is entitled to have been elsewhere this week, but the dynamic left hander has displayed a steely focus throughout and rallied in terrific style in this last-eight clash to win each of the last three frames having slipped 4-2 behind.
After the first four frames were shared, Carter appeared to have taken control of the match when putting together back-to-back breaks of 86 and 53 to move within a frame of victory.
However, he was chained to his chair thereafter, failing to register a single point as Lisowski produced a devastating burst that powered him into his first semi-final of the season.
After dominating frame seven to close to within one, Lisowski suddenly clicked into gear with a fluent run of 82 before producing a magnificent total of clearance of 135 to secure victory.
Lisowski will now face Joe Perry who beat Ricky Walden 5-2 in his quarter-final.
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