Davies and Daly trailed Pak and Azinger by five strokes entering the final round, but the pair combined to return a seven-under-par 64 in the alternate-shot format to finish at 24-under-par 260, level with Pak and Azinger, who carded a 69.
Davies made the winning play-off putt on the par-three 17th after the teams had parred the first two play-off holes "That's the longest putt I've holed all year," said Davies, whose short-game problems contributed to an LPGA season bereft of a victory.
"John helped me with my confidence, telling me I'm rolling it well. To actually have someone else who is hitting all the shots with you, I was cheering his shots on as much as I was cheering any time I made a putt."
For the embattled Daly, the win was the bright spot in a year that saw him admit to a relapse to his drinking and gambling. "I've been working for the last three months very, very hard on my game," Daly said. "I've just got to keep working at it."
Davies sank the critical putts to maintain their challenge and Daly said: "She made three in the last four holes. That was just awesome."
Colin Montgomerie has won the European Tour's Golfer of the Year award for a record-breaking fourth time. Winner of the Order of Merit for a seventh successive season - another record - Montgomerie regains the trophy he lost to Lee Westwood last year.
Seve Ballesteros and Nick Faldo are both three-time winners of the Golfer of the Year award, while Bernhard Langer has taken it twice.