Brown's win against the Big Green, coupled with Cornell's 15-10 victory against Princeton, gives the Bears a share of the Ivy League Championship and the number one seed in the Ivy League Tournament, which will be held at Brown's Stevenson Field on May 1 and 3.
Brown finishes with an overall record of 12-3, 4-2 in the Ivy League.
Molloy had five goals and two assists in the win. He now sits in second place in program history in single-season goals with 58.
Brown also received a strong showing from Kylor Bellistri, who netted three goals, and Henry Blynn and Brendan Caputo, who netted two goals each.
Bears goalie Jack Kelly was brilliant once again, making 10 of his 15 saves in the opening 30 minutes.
Dartmouth was paced by Jack Korzelius, Phil Hession and Adam Fishman, who scored two goals each.
Goals by Caputo and Molloy gave the Bears a 2-0 lead early in the opening quarter, but the Big Green came back to tie the game at 2-2 on scores by Korzelius and Hession midway through the quarter.
Blynn's goal gave Brown a 3-2 lead but Fishman quickly responded for Dartmouth to tie the game.
Molloy's goal with 2:37 remaining in the opening quarter gave Brown the lead for good at 4-3 after one quarter.