The Orange got out to a 4-1 lead on goals by Rice and Henry Schoonmaker, but the Big Red (9-2) responded with a 4-0 run to take its only lead of the game at 5-4.
Syracuse scored the next two goals to make the score 6-5 at halftime.
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The Orange pulled away in the second half as Staats scored four straight goals to make the score 12-6 with 9:44 left.
"We knew how talented [Staats] was, but I give him a lot of credit. He scored some really nice goals tonight," Cornell interim coach Matt Kerwick said.
Goalkeepers Dominic Lamolinara (St.Mary's) and Bobby Wardwell combined for 12 saves for the visiting Orange.
Dan Litner led the Big Red with three goals in a losing effort.
Midfielders Trevor Bosco and Jim Marlatt (River Hill) each recorded a hat trick for Notre Dame, which ended a two-game losing streak.
"We didn't have a lot of energy and we didn't seem to have our legs, but we found a way to win and we feel good about that," Fighting Irish coach Kevin Corrigan said.
No. 16 Hofstra 15, Marist 10: Torin Varn had six goals on 14 shots as the host Pride (8-3) never trailed in a victory over the Red Foxes (3-8).
Marist's Drew Nesmith also had six goals in the loss.
No. 18 Princeton 10, Lehigh 9, 2OT: Tom Schreiber had four goals, including the game-winner in the second overtime, as the host Tigers (6-4) defeated the Mountain Hawks (9-4).
Princeton's Kip Orban tied the game with 2.7 seconds remaining, sending the game into overtime. Princeton outshot Lehigh 52-31, but neither team ever led by more than one. The Mountain Hawks' three-game winning streak ended with the loss.
"There was good energy on both sides of the ball," Lehigh coach Kevin Cassese said. "Both teams really wanted this game. Both teams fought tooth and nail for it, and that's how you get a double-overtime game."
Salisbury 25, Wesley 6: The host Sea Gulls (12-1, 5-1 Capital Athletic Conference) went on an 11-0 run that included three goals in the first five minutes of the game in a rout of the Wolverines (5-6, 0-5).
Mike Kane and Thomas Cirillo each had four goals for Salisbury.
Neumann 17, Notre Dame (Md.) 8: Mandy Fedak and Rebecca Dolenti each had a hat trick as the host Knights (8-3, 4-0 Colonial States Athletic Conference) defeated the Gators (2-8, 1-5).
Sawyer Butto had a team-high four goals for Notre Dame of Maryland.
Salisbury 18, Christopher Newport 10: Bethany Baer (Westminster) had four goals to lead the Sea Gulls (13-0, 5-0 CAC) past the host Captains (7-6, 2-3).
Salisbury opened with a 4-0 run and led 10-5 at the half. Ashton Marshall had four goals for Christopher Newport.
No. 12 Penn State 13, Cornell 8: Madison Cyr (Winters Mill) had a career-high five goals to lead the Nittany Lions (8-5) over the visiting Big Red (5-6).
Penn State outshot Cornell 28-16. Lindsay Toppe led the Big Red with three goals.