Trailing 70-61 with 3:32 left, Cornell made 14 of 18 free throws and its lone field-goal attempt, while Dartmouth shot 1 of 7 from the field and did not register a free throw in that span.
Darryl Smith made two free throws with 1:14 remaining to tie and take the lead for Cornell (9-9, 2-2 Ivy League).
Evan Boudreaux had 24 points and 16 rebounds for Dartmouth (7-11, 1-3), and Connor Boehm added 22 points.
Cornell shot 63 percent in the first half en route to a 41-35 halftime lead. But the Big Green outscored Cornell 23-5 to start the second half.
Morgan's 23 free-throw attempts are third in school history, all coming in the second half - 19 in the final 8:53.