Godbout , who scored the team's last nine points, made only 5-of-14 shots from the field, but connected on 14-of-15 shots from the foul line.
The Golden Bears, who clinched a home game in the GNAC Tournament that begins on Tuesday, improves to to 17-7 with their seventh consecutive victory and 13-4 in the conference.
JWU (13-11, 11-6) led 33-31 at half-time and increased its margin to 43-36 with 16:12 left in the game.
The Golden Bears pulled within 45-44 at the 12:17 mark, before the Wildcats went on a 10-2 run for a 55-46 advantage following a basket by junior forward Conor Moran (Rio de Janeiro, Brazil) with 7:56 remaining.
Western New England College responded with an 11-0 run over a 5:41 span for a 57-56 lead following two free throws by Godbout with 2:45 left.
Senior guard Brian Bannister (Boston, MA) put the Wildcats ahead for the last time when he nailed a three-pointer for a 58-57 lead 30 seconds later.
Godbout made two free throws and scored on a lay-up to give the Golden Bears a 61-58 lead with 1:14 remaining.
Bannister answered with a basket to pull JWU within 61-60 with 58 seconds left.
After Godbout missed a jumper with 27 seconds remaining, the Wildcats had a chance to go ahead ahead but senior guard Daisuke Yoshimotto (Osaka, Japan) missed a wide open three-pointer with 7.1 seconds remaining.
Godbout then made two more free throws with 6.9 seconds left.
Johnson & Wales quickly moved the ball up court and called time-out with 4.1 seconds. Yoshimotto then missed another three-pointer at the buzzer.
Sophomore forward Jim Bayne (Bristol, CT) added 12 points and nine rebounds for the Golden Bears while senior center Mike Cutler (Amherst, MA) contributed 11 points (4-of-5 from the field) off the bench.
Freshman point guard Asif Abdul-Wadud (Springfield, MA) played the entire game (as did Godbout) finishing with seven points, six rebounds, and three assists in only his second varsity start.
Bannister led the Wildcats with 20 points on 9-of-16 shooting from the field, including both three-point attempts. He also had eight rebounds and four steals. Moran finished with 10 points and five rebounds before fouling out.
JWU defeated WNEC by the score of 67-60 January 15 in Providence, Rhode Island.