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Trump produced the tournament's highest break of 140 to lead 5-4 and, after Robertson had claimed the 10th frame, won the decider with a 129.
The match was hailed as one of the greatest in Masters history by BBC commentator John Virgo.
World champion Stuart Bingham ended the challenge of John Higgins as he secured his progress with a 6-3 win.
Bingham will now play Ronnie O'Sullivan in the semi-finals, with the match starting at 19:00 GMT on Saturday.
Higgins made some unexpected mistakes which allowed Bingham to capitalise in a match which failed to match the quality of the contest between Trump and Robertson.
Robertson, ranked three in the world, hit a 139 as he recovered from 2-0 down to level at 2-2 against Trump.
Breaks of 105 and 78 helped Trump go two frames in front again, only for Robertson to once more restore parity, before Trump went on to secure victory.
"What a fantastic exhibition of snooker. One of the greatest games we have seen in Masters history, and any other tournament I've seen," said former UK Championship winner Virgo.
Trump will play Barry Hawkins at the Alexandra Palace in London in a semi-final match which starts at 13:00 GMT on Saturday.
The match-winning break started with a superb red which was cut into the middle pocket, with six-time world champion Steve Davis describing the piece of opportunism as a "chance from nothing".
England's Trump punched the air after sinking a blue, which sealed the win against Australia's UK Championship winner Robertson.
"In the first round I was really nervous. I was looking for form and when I was out there today knew I was playing well but Neil kept coming back at me," said Trump.
"As soon as that [opening red in the final frame] went in I thought don't mess up.
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"If ever I am the underdog I seem to play well and whenever I am the favourite I put too much pressure on myself.
"Knowing me I could go out and play terribly in the next round."
13:00 Barry Hawkins (Eng) v Judd Trump (Eng)
19:00 Ronnie O'Sullivan (Eng) v Stuart Bingham (Eng)
13:00 and 19:00
World number five Judd Trump beat Neil Robertson 6-5 in a Masters quarter-final that featured six century breaks.