The 24-year-old Northern Irishman lost the opening two frames to Davis but rallied well to go into the mid-session interval level at 2-2.
Allen, a semi-finalist at the World Snooker Championship last year, edged ahead for the first time and then had an outstanding chance of a 147 maximum in the sixth frame.
After six reds he lost position by a fraction and had to take pink rather than black, but as if to prove a point to himself he went after the 146 and brilliantly reached his target.
Whereas a maximum would have brought him prize money of 157,000, Allen had to settle for being the frontrunner for the highest break prize, which pays 10,000.
Although 146 breaks are more rare than 147s, Allen has made two, with his first having come in qualifying for the UK Championship in 2007. There have been nine maximum breaks in World Championship history, with Stephen Hendry recording the most recent during his quarter-final against Shaun Murphy at last season's tournament.
Allen finished the session with a 5-3 lead, Davis winning the eighth frame to stay in touch ahead of today's resumption.
Murphy set up a second-round meeting with China's Ding Junhui by scrambling over the line against Gerard Greene.
He almost made a complete mess of the session though, and was clearly relieved to polish off a 10-7 win. Greene trailed 8-1 overnight and Murphy soon made it 9-2. But Greene managed to win six of the first seven frames of the session. The Kent-based Northern Ireland left-hander had Murphy in a touch of trouble when he fired in a break of 92 to narrow the gap to 9-7. But a chance came Murphy's way early in the 17th frame and he calmly recorded a break of 87.
Meanwhile, 2002 world champion Peter Ebdon is leading resistance to promoter Barry Hearn's proposed takeover of the business behind the sport.
Hearn plans to take over World Snooker and turn the sport into a multi-million-pound roadshow with weekly tournaments and, eventually, winnings to rival golf's huge prizes. His Matchroom business would take a 51 per cent share, and leave the rest in the hands of players and fellow promoters.
Ronnie O'Sullivan, John Higgins, Neil Robertson and many other leading names have given their full support to the man who has made darts fashionable. But Ebdon countered: "The players need to understand this is not for a year, this is not for two years, this is not for five years, this is not for ten years. This is it, forever, gone. You can't possibly give control of the association away under any circumstances."