World number two Trump ruthlessly mounted a 4-0 lead going into the mid-session interval, leaving him requiring one more to see off the eight-times world billiards champion from India.
But Advani, who took two major scalps in the form of 2005 world champion Shaun Murphy and 2006 winner Graeme Dott to become his nation's first-ever quarter-finalist in a ranking event, attempted a fightback with two frames on the trot bounce after the break.
Trump then closed out the win at the third attempt to set up a last-four meeting on Saturday with the winner of Friday evening's all-Scottish quarter-final Stephen Maguire, following his 5-3 win over Alan McManus.
The Bristolian picked off his opponent with multiple small breaks over the opening pair of frames for a 2-0 lead before runs to 52 and 78 put him on the verge of a whitewash in short order.
His early play hindered seemingly by nerves, the 4-0 deficit allowed Advani to play with nothing to lose and he managed to pull one back before taking another after Trump had missed frame ball on a knock of 59, completing a fantastic clearance to steal for survival.
Trump regrouped to hit 69 and 29, with Advani missing a chance to attempt another massive come-from-behind in between, to avoid an upset and reach another ranking semi-final.
The other quarter-final of the session, which saw Stuart Bingham overcome Ken Doherty 5-3, was far less swift as the two engaged in prolonged battles for the majority of the frames before the Englishman finally pulled away from the 1997 world champion.
Doherty began emphatically enough, rounding up an 88 break to win the opening frame before doubling his lead soon after. It was Bingham who proved more resolute in safety exchanges, however, and combined with smooth cueing he was able to peg the Irishman back to parity at intermission.
Doherty took an arduous fifth frame with excellent snookering only for a 52 to inspire Bingham to level once more, and the 36-year-old from Basildon made it three frames on the bounce with a well-timed rise to the ascendancy, finished with a difficult 71 to seal success.
Up next for Bingham will be defending champion Ding Junhui, who made short work of Robert Milkins.
Ding started with an even 50 half-ton before edging the second frame for a 2-0 lead, which doubled after ruthless runs to 104 and 74.
Milkins finally showed off the snooker which got him to the last eight with a 108 after the break to dodge a 'bagel' scoreline but despite racking up a 52 in frame six, could only watch after an error allowed Ding to claw back and steal the frame and the match with a 63 clearance.