The host Irish (3-0) scored two goals in the last minute of regulation to force overtime and kept the Tar Heels (2-2) off the scoreboard in the final 16 minutes, 44seconds of the game.
"You leave the field with a big smile on your face after a game like today," Notre Dame coach Kevin Corrigan said.
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Kavanagh scored four goals, including the tying goal with 11.8 seconds left in regulation and the game-winner. He also had two assists.
Notre Dame led 5-4 at halftime, but North Carolina scored four straight goals in the second half to take an 8-5 lead.
The Tar Heels were up 9-6 with 8:29 left in regulation. The Irish rallied with goals by Sean Rogers and Kavanagh to send the game into overtime.
No. 2 Cornell 19, No. 6 Colgate 3: The Big Red (3-0) seized an 11-0 lead and cruised to a win over the visiting Raiders (3-1).
Rob Pannell had two goals and five assists for Cornell. Colgate's Peter Baum took three shots and was held without a goal for the first time in 21 games dating to 2011.
Senior goalie AJ Fiore made 15 saves behind the Big Red's defense, which limited the Raiders to 11 shots in the second half.
Brown 9, No. 7 Massachusetts 8, OT: Despite winning only three of 20 faceoffs, the Bears (1-1) defeated the host Minutemen (3-1).
Brown scored three goals in four minutes to tie the game in the fourth quarter, and Daniel Mellynchuk scored the game-winner less than two minutes into overtime.
Penn State had an 8-2 lead at halftime, but Ohio State rallied with six straight goals. After the Buckeyes took a 9-8 lead, Shane Sturgis tied the game with 4:04 left in regulation.
No. 10 Denver 15, No. 13 Penn 12: Led by Wes Berg's game-high seven goals, the host Pioneers (4-1) defeated the Quakers (2-1).
Berg had a hat trick in 37 seconds in the first quarter and scored six first-half goals.
Penn outshot Denver 46-34, but Pioneers goalie Jamie Faus made eight saves after being inserted at halftime.
No. 15 St. John's 12, Siena 11: Colin Keegan's fifth goal of the day was the game winner as the visiting Red Storm (3-1) defeated the Saints (1-3).
Kyle Curry scored with 11:52 remaining to give Siena a 10-9 lead, but St. John's responded with two goals in 15 seconds.
The Saints' Colin Clive completed a hat trick and tied the game with 6:05 left, but Keegan scored the game-winner less than three minutes later.
No. 16 Yale 15, No. 18 Albany 8: The Bulldogs (1-1) beat the host Great Danes (1-2) behind Conrad Oberbeck's six goals.