The Great Danes (11-12, 4-4 America East Conference) trailed 37-22 with 2:16 left in the first half and 39-27 at the break.
The Retrievers (3-18, 2-6) still led 49-36 with 14:24 to go when Albany went on a 14-2 run to get within a point.
The Great Danes tied it at 55 on an Ambrose 3-pointer with 7:47 left, but didn't pull ahead until Mike Black's layup made it 64-63 with 23 seconds left.
Aronhalt's two free throws with 2 seconds left gave the Great Danes their third straight win.
Albany shot 38.5 percent from the field (25 of 65) while UMBC shot 39.7 percent (25 of 63).