Robertson was never ahead until the 11th frame in a turgid contest that flattered to deceive with neither man producing their best form.
Allen should probably have ousted the world number one having lost 6-5 to the man from Melbourne in last year's quarter-finals.
The Northern Irish player will reflect on breaking down on 30 in the final frame when he ran out of position on a red from potting a pink before leaving himself without a pot on the next colour.
Expecting a re-rack holding a 30-13 lead, Allen played a careless safety shot giving Robertson the opportunity to sink a red from mid-range that he accepted with some glee before contributing 59 to seal his progress.
UK champion Robertson will face Stephen Maguire in the last eight on Friday night with Robertson hoping conditions are cooler in mid-January at the usually airy London venue.
"The playing conditions were far too hot to play in," said Robertson. "They're talking about the heat at the Australian Open tennis in Melbourne at the moment, but that was just ridiculous.
"Temperatures were too hot. The cue ball played awful for the first four frames, and we had to get it changed.
"The match didn't really start until 2-2, and even then we were guessing what might happen.
"It needs to be lowered a few degrees so the tables play a bit better
"A lot of people in the crowd were complaining about it being too hot in the arena.
"You almost don't want to stick your back against the chair because you are sitting there with a shirt and waistcoat on. It is very uncomfortable."
Robertson is of a mind that snooker players should be allowed to test out the match table one hour before they get underway.
"Perhaps if you get here an hour earlier, and have a hit on the practice table it would be better so you know what to expect," said Robertson. "If it was early enough, I wouldn't have any problem with that.
"Mark had chances to pull away. He should have been 5-3 and 3-1 up. He was unlucky in the final frame, but he shouldn't have been out of position a shot earlier.
"He will be absolutely sick about that."
For the record, Robertson made breaks of 46, 101, 41 and his match-winning 59.
Allen looked the likelier player but runs of 114, 65 and 52 will mean little in the end having led by one frame until Robertson broke the pattern of trading frames by winning the final two.
"I'm disappointed with the way I played," said Allen. "It was such a careless shot to leave Neil that red in the final frame. He showed why he is world number one by taking that chance.
"Both of us were well below par. I came here with good form, but that was a bad day at the office.
"I come here expecting to win these tournaments. That is the pressure I put on myself. I'm probably trying to hard."