Calby's second goal of the day at Pape Field marked the sixth and final lead change of the back-and-forth contest as the Big Green (1-0) were able to hold off the Ducks (0-1) in the final minute of the extra period.
The junior's game-winner came on Dartmouth's lone shot of overtime.
In addition to Calby's multi-goal effort, senior Hana Bowers scored three times and assisted on the winner, notching her first two tallies of the season just a minute apart in the first half.
Trailing 7-6 in the final 10 minutes of regulation, sophomore Frances Bird stepped up and delivered the equalizer with 9:08 remaining, eventually helping the visitors force the overtime session.
An All-Ivy Second Team goalkeeper from a season ago, junior Kristen Giovanniello made six saves to earn her first victory of the year and the 24th of her career (24-10). Her one save in the first OT period proved to be the only stop by either team's goalkeepers following the conclusion of the second half.
At the other end of the field, Oregon's Caroline Federighi stopped five shot, while allowing eight in being saddled with the loss.
Lindsey Allard and Jaclyn Leto also contributed to the scoring for the Green as each added one goal apiece in the winning effort.
For the Ducks, Nikki Puszcz and Sam Kiser each found the back of the net twice, while Shannon Propst added a goal and two assists.
Dartmouth was outshout by the home team, 20-19, but managed to earn the win in just the second all-time meeting between the two programs. Oregon came to Hanover during the 2011 season and fell to the Big Green, 12-10, in the first match-up.
The Green and White are back in action on Wednesday afternoon as they open their 2013 home schedule with in-state rival New Hampshire at Scully-Fahey Field. The opening draw between the two Granite State squads is set for 3 p.m.