Robertson came to the table in all sorts of trouble early in the final frame after Allen had left his opponent snookered and apparently poised for defeat before an expectant crowd at the London venue.
But 2010 world champion Robertson - who ousted Ding Junhui 6-5 by winning the final three frames in the first round - managed to escape from the snooker behind the yellow to go up and down the table and hole the red that Allen had left over a top pocket.
Robertson was fortunate in landing on the black, but there was nothing lucky about the 105, his third century of the match, that secured a meeting with John Higgins or Shaun Murphy in the semi-finals.
"In terms of judgement and under the circumstances, it was definitely one of the best shots I've ever played..probably the best," said Robertson.
"Obviously, there is an element of luck in potting it so cleanly to end up on the black, but I had to give myself every chance to end up on the black. I could have played it at pace and misjudged it..and opened up all the reds.
"It is lucky, but not a fluke. It was kind of like reading a horrible putt in golf that goes in.
"I felt sorry for Mark to go out of the tournament without doing anything wrong."
For the record, Allen made 49, 138, 72, 68, 60 with Robertson running in 66, 73, 111, 101 and 105 after an epic tussle.
Robertson admits he has learned lessons from last year despite winning the event. Judd Trump was left sporting a hangdog look and complained about Robertson's "slow play" after losing in the last four, but Robertson concedes Trump may have had a valid point.
"There was obviously a lot said between me and Judd after last year's match," said Robertson.
"I'm playing similar to last year scoring-wise, but trying to play a little bit quicker when I'm in the balls. Judd was probably right in a way that I probably got myself bogged down a little bit too much.
"I'm not doing it on purpose to put off the other guy, but perhaps there is pressure or not having the confidence to go on and attack.
"I'm trying to get on with it when I get in the balls to try it and make it a bit better so people can enjoy it a bit more. If all the players can maybe just quicken up a bit, we can create some more great matches.
"I played too slow against Judd in (losing) the semi-final of the (2011) UK Championship and then played some great matchplay snooker against him here, but I think he is right..I should play my natural game more.
Robertson continued: "I'm never going to play as quickly as Judd or Mark Allen, but I also don't think I should get bogged down with safety too much."
Allen had enjoyed a 6-4 win over Robertson in the Masters quarter-finals two years ago at Wembley Arena - which remains his last success against the Melbournian - with Robertson completing a 6-3 victory in the first round on his way to winning last year's event.
Perhaps unsurprisingly, the first four frames galloped past in less than an hour with both men producing some bewitching stuff.
Robertson pocketed the opening frame with a lovely run of 66 after Allen saw a red struggle out on 20 with a bad miss only for the Northern Irishman to restore parity aided by a knock of 49 in the second frame.
The quality snooker continued as Robertson ran in a quite immaculate break of 73 that never looked on when he came to the table.
Robertson barely had time to revel in his contribution and a 2-1 lead before he had to sit in his chair and watch Allen piece together 138 - an effort that overtook his 136 highest break of the tournament set against Mark Davis on Sunday.
Allen continued where he left off after the mid-session interval as 72 saw the Antrim man move 3-2 ahead.
Robertson's response was terrific as he pieced together 111 to level at 3-3 as the one-visit nature of the afternoon continued at a glorious pace.
The first scrappy frame was won by Robertson after some uncharacteristically stuttering play by both men as the Australian edged 4-3 clear, but Allen immediately responded with 68 to level at 4-4.
With neither man wilting when presented with the slightest of opportunities, Robertson held himself together in running in a 101 to move one frame from the last four.
It was of minimal surprise to see Allen level at 5-5 boosted by a run of 60 in the tenth frame as the top seed Robertson broke down on 39 in trying to respond, but he was destined to save his best for last.
"It was a great hit by Neil, but to pot the red and end up on the black at the right angle is quite fortunate," said Allen. "But Neil apologised. Hopefully, one day it will happen for me.
"I think my safety play cost me today. It was nowhere as good as Neil's. Against Neil, who in my opinion is the best player in the world, it has to be a lot better than that.
"I scored well and didn't miss many balls, but unfortunately I didn't get the chance I was looking for in the final frame.
"People ask why I don't win more in snooker, perhaps that is why. If Neil shouted out "You beauty" in the final frame, I would have cracked up. But he is a good sportsman and apologised at the end.
"He also only fluked one ball, he still had to go and pot the rest. Not many players would have made a century from that position."