The Midshipmen were trying to run out the clock, and a failed pitch on their option offense turned into a fumble recovery by the Eagles' Jolonn Dunbar at the Navy 40-yard line with 1:43 remaining.
A 15-yard pass by Matt Ryan to Ryan Purvis and two rushes set up Aponavicius' winning kick and gave Boston College its seventh straight bowl win.
Ryan passed for 242 yards with a touchdown and also ran for a score for the Eagles (10-3), who haven't lost a post-season game since a 62-28 setback against Colorado in the 1999 Insight.com Bowl.
Boston College posted its first 10-win season since 1984, when it was led by Heisman Trophy winner Doug Flutie.
Shun White ran for 116 yards on seven carries for Navy (9-4), which out-gained Boston College 403-315. The Midshipmen had won their previous two bowl games.