Trump was forced to hold off a late Robertson comeback, but the 32-year-old ultimately sealed a 5-3 victory in the quarter-final.
The 2019 world champion raced to a 4-1 lead and could have secured an early win had he potted a red into the middle pocket. However, a miss from Trump allowed his Australian opponent to return to the table and make a frame-winning break of 46.
The current world number four continued his newly-found momentum into the next frame and prevailed once more to go within one of his rival.
Just as the 40-year-old was about to level the match in the eighth frame, a miss on the black allowed Trump to finally close out the match.
Trump told Eurosport: "I started OK, Neil was missing a couple and I was able to pick up the pieces. I had a good chance to win it 5-1 but I got a bit panicky. In the end I needed him to gift it to me, really."
The Juddernaut will battle Iran's Hossein Vafaei in the semi-final. This year's Snooker Shootout winner bagged his place in the final four following a 5-4 win over China's Zhang Anda.
Elsewhere, Jack Lisowski booked his spot in the semi-final following a 5-4 victory over Ali Carter.
The 30-year-old, who is still chasing his first ranking tournament title, fought back from two frames down to bag an impressive win.
Lisowski will now take on Joe Perry for a place in the final after The Gentleman made easy work of Ricky Walden, winning 5-2.