Greaves, coming off an 86-match winning streak in the women's game, opened by holding her throw with tops on her first attempt.
Gurney responded by holding his throw, then broke Greaves after she suffered an unfortunate bust. He had an opportunity to win the set, but missed a double, allowing Greaves to finish 40 and level the legs at 2-2.
Although Greaves had the darts, Gurney reached a finish first and secured tops to take a 1-0 lead in sets.
Gurney began the second set with the match's first maximum and finished 54 to gain control. Greaves responded with a 15-darter and began to build momentum.
A fine 128 finish on the bullseye saw the 21-year-old break Gurney's throw in 12 darts, before a spectacular 10-darter brought back parity to the scores with an excellent 108.75 average for the set.
Both players held their throws to start the third set. Gurney then broke with a 41 finish on double 16 to take the lead. Greaves responded with a 13-dart break to force a decider, but a missed dart at tops allowed Gurney to move within one set of victory.
Greaves impresses
Seeking to become only the second woman to win a match at Ally Pally after Fallon Sherrock, Greaves broke again in the first leg of the fourth set. She held with double 18 in the second leg and finished tops to force a decider.
Greaves had the darts in the fifth and final set and was outperforming Gurney. Another 13-darter, finished on double 10, put Gurney under significant pressure. Gurney managed to hold and level the set at 1-1.
The third leg was closely contested, but a phenomenal 144 checkout - seemingly from nowhere - allowed Gurney to throw for the match.
A late slip from Greaves gave six potential match darts for Gurney, but after missing four match darts, Gurney would finally get over the line.
Greaves won more legs, landed more 180s and had a higher average (91.55 to Gurney's 89.77), but Gurney's timely 144 finish secured his place in the next round.
'That woman has some cajones'
"She's the best women's darts player on the planet," Gurney told Sky Sports.
"If there's going to be a darts player that can win the World Championship or any majors, that's the player.
"Even from a man's point of view, that woman has some cajones. What a dart player, honestly what a dart player.
"Everyone said she was the favourite coming into the game, but sometimes I show my class and my composure, and I got over the finish line.
"The only thing that separated us was the 144.
"I missed some doubles, but fair play to Beau. All this crowd should respect her because next year she's going to be a force to reckon with.
"She's going to win some PDC tournaments."
Gurney will face Callum Rydz in the next round.
Aspinall reels in big fish
Fresh off winning the Players Championship Finals, Nathan Aspinall got his World Championship campaign up and running with a nervy 3-1 win over Lourence Ilagan.
Aspinall roared into the contest to his usual soundtrack of Mr Brightside, and swiftly began peppering the treble bed and holding throw in 15 darts.
Ilagan responded with a maximum of his own and, amid laughter from the commentary box and an ill-timed appearance from the now infamous Ally Pally wasp, settled into a rhythm that Aspinall struggled to disrupt.
A stunning 177 took Ilagan to the brink and, despite missed chances and a failed 142 from Aspinall, the Filipino kept his nerve to nail double 10 and steal the first set.
The Asp responded with an immediate break via a 114 checkout, paving the way to clinching a comprehensive second set to level the scores, upping the ante with a 95.75 average.
Both players held their legs before Ilagan took out a vital 64 to break and restore his lead, but a bullseye finish on 84 swung the pendulum back in the Englishman's favour. A 14-dart leg finished on tops eventually put Aspinall 2-1 up in sets.
The darts would again go with the throw in the fourth set.
Aspinall's average began to dip, while Ilagan at one point averaged 102.5 in the set. An intriguing 16, 16, double 16 to take out 64 put Ilagan 2-1 up in legs, and combined with poor scoring on his throw in the next, suddenly the two-time semi-finalist was staring down the barrel of an undesirable decider.
Still, missed darts opened up a chance at 16 to level the set once more, which Aspinall took.
Needing a break for the match, Aspinall left himself 170, and the clutch finish came at the perfect moment to see Aspinall set up a second-round clash with Leonard Gates, averaging a smidgen over 94.
O'Connor hits highest average so far
In the opening encounter of the evening, William O'Connor steamrolled Poland's Krzysztof Kciuk with a dominant 3-0 victory.
An impressive 102.36 average, the highest of the tournament so far, plus a 53 per cent success rate on the doubles and seven 180s, set the tone for the evening, as the Irishman put in one of the performances of his career to set up a second-round clash with the mighty Michael van Gerwen.
"I feel like I've got more to give," O'Connor told Sky Sports afterwards.
"No matter how well you're playing, you always feel like you can play a little bit better.
"I'm very happy with how my doubles went but I know my scoring can go a little bit better, but at the same time I'm very happy with my performance.
"The next game is going to be very, very hard, and I know I'm going to have to play at least like that, if not better - I can't have those slack darts."
In the final match of the round, Ireland's Keane Barry stormed past Tim Pusey with a straight-sets victory.
Earlier in the day, Northern Ireland's Josh Rock stormed into the second round of the PDC World Darts Championship, rounding off the Friday afternoon session with a hard-fought 3-1 victory over a game Gemma Hayter.
Rock averaged 90.18 for the match, hitting three 180s. Both players hovered around finishing a third of their doubles.
There were also wins for Kevin Doets, Ryan Meikle and Leonard Gates.
Friday evening session results:
William O'Connor 3-0 Krzysztof Kciuk
Nathan Aspinall 3-1 Lourence Ilagan