The Big Green (5-12, 1-2 Ivy League), who also snapped a 10-game losing streak to the Big Red, led by 28 points with 12:01 to go, but saw the lead dwindle to single digits in the final minute.
After Dartmouth used a 21-4 run to lead 31-14 at halftime, Cornell (4-13, 0-3) scored only six points in the first 8 minutes of the second half.
Leading 48-20, the Big Green hit a slump as the Big Red closed to 56-46 with 2:21 to go, then 61-53 with 52 seconds left.
Griffin made two free throws, then Chris Wroblewski's two jumpers got Cornell within six with 16 seconds to go.
Rufful hit a foul shot and the Big Red missed two 3-pointers in the last 13 seconds.
Errick Peck scored 20 to pace Cornell, which has lost three in a row.