Todd added two assists for the second-seeded UNB, while Antoine Houde-Caron had a goal late in the third period. Daniel LaCosta made 13 saves for the win.
"In this format, winning game one is crucial. This tournament is a best-of-one, really," said UNB head coach Gardiner MacDougall, who hopes to lead his program to a fourth University Cup title in seven years. "We showed a lot of character tonight in a challenging environment. I thought we played a tentative first period but when they scored, it kind of woke us up."
Jimmy Bubnick was the lone scorer for Saskatchewan, while Ryan Holfeld stopped 26 shots.
The Varsity Reds were 2 for 5 on the power play, while the Huskies went 1 for 4 with the man advantage.
Bubnick's goal gave Saskatchewan a 1-0 lead, but Todd tied it 1-1 within the next six minutes.
Carroll, who led the Atlantic conference in the regular season with six game-winning goals, broke the deadlock with 57 seconds left in the second. Well positioned in front of the net, the sophomore forward from Strathroy, Ont., one-timed a nifty pass from Neil MacNeil over Holfeld's glove to complete a perfect three-way passing play.
UNB takes the early lead in Pool B and will rest until Saturday, when the Varsity Reds face No. 3 UQTR in their second and last game of the preliminary round.
Fifth-ranked Saskatchewan takes on UQTR on Friday afternoon.
Earlier Thursday, Keaton Hartigan made 42 saves as the No. 6 Waterloo Warriors shocked the top-ranked Alberta Golden Bears 2-1.
Justin Larson and Blake Chartier scored power-play goals for the Warriors, who were outshot 43-13 by Alberta. Colin Behenna assisted on both goals.
Torrie Dyck scored a power-play goal for Alberta with 3:30 left in the third period, but the Golden Bears were unable to put an equalizer past Hartigan.
Hartigan was particularly impressive in the second when he turned aside 22 shots.
"Keaton has been the heart and soul of our team during our playoff run and he's one of the main reasons we’re here," said Waterloo head coach Brian Bourque, whose troops settled for sixth place in the OUA East in the regular season with a sub-par 12-11-5 record but made an improbable run to the conference final in the playoffs.
Unsurprisingly, Hartigan was named Waterloo's player of the game.
"I just felt comfortable out there," said Hartigan, a fifth-year senior from Kitchener, Ont. "We allowed a lot of shots but I got a good look at most of them, and the guys did a great job at clearing rebounds."
CIS goalie of the year Kurtis Mucha made 11 saves for Alberta.
With the win, the Warriors take the early lead in Pool A and get to rest until Saturday afternoon, when they'll face No. 4 Saint Mary's.
The Golden Bears, celebrating their 100th year on the ice, play Saint Mary's on Friday afternoon. They will need to win by at least two goals and hope the Huskies beats Waterloo in order to advance to the gold medal game.
Larson opened the scoring at 14:09 of the first period on a power play when he one-timed a nifty pass from Behenna past Mucha.
After surviving a second period in which they were outshot 22-4, the Warriors doubled their lead 7:56 into the third. Blake Chartier took another nice feed from Behenna and scored the Warriors' second power-play goal of the afternoon.
Dyck gave Alberta some life late in the game but it was not enough.
Later Thursday, UNB faced host Saskatchewan in Pool B action.