The potentially mouthwatering meeting between Allen and Robertson on Thursday will be their third at the 16-man invitational tournament over the past three years.
Allen enjoyed a 6-4 win over Robertson in the 2011 quarter-finals at Wembley Arena with the Australian completing a 6-3 victory in the first round on his way to winning last year's event.
Robertson progressed earlier in the day with a 6-5 win over Ding Junhui.
"I'm ready to win any tournament. My results of late haven't been great, but I'm just glad to get over the line," said Allen.
"I seen the end of Neil's win over Ding earlier today. Ding was a bit unfortunate not to split the reds to win the match, but Neil showed what a great player he is and how much bottle he has to come through.
"Hopefully, I won't give him the same chances against me. We always have pretty close matches, but it is up to me to go out there and play my own game."
Allen made a brilliant start to the evening against Davis as he watched his opponent break off before rolling in a long red and compiling an absorbing knock of 136.
Hastings potter Davis - making only his second appearance at the tournament a decade after he lost 6-5 to Alan McManus in the first round - responded with 62 to level at 1-1 only for Allen to dominate a scrappy third frame in moving 2-1 clear. A couple of efforts in 40s saw Davis level at 2-2.
Davis missed a tough red to a middle in the fifth frame after Allen had himself failed to slot an ambitious cut on a red to a middle bag. With the reds wide open, Allen compiled 79 for a 3-2 advantage.
The sixth frame found Davis wanting again as he ran out of position after Allen had missed a black off the spot at pace. Allen pieced together 42 before a terrific pot on a red to a middle hole handed him a two-frame lead for the first time in the match.
Davis had chances to claw back to 4-3 in the next frame, but he broke down on 35 in jawing a pink before failing to hole the tricky final red at pace along the top cushion. Allen cleared to the pink to move within one frame of victory.
Another Davis error early in the eighth frame enabled Allen to compile an 80 to complete the victory.
"I missed one or two tricky ones after the interval," said Davis. "You can't afford to do that with Mark hitting the ball so well. I enjoyed it out there, but it wasn't to be.
"He played well. I was delighted to be here, but hope to get back next year."