Goode made 1 of 2 free throws 14 seconds into overtime and added two at the 3:26 mark to put the Mountaineers (3-2) ahead 57-54. Cajou's 3-pointer with 2:38 left made it 60-54.
The Midshipmen (2-4), who shot 0-for-6 in overtime, never led and got their only tie with 22 seconds left in regulation when Cajou fouled Romeo Garcia, who made both free throws.
Navy trailed 32-23 at the half but a 15-7 run pulled the Midshipmen within 39-38 with 11:53 left. Chris Harris scored nine of his 17 points in that stretch, including two free throws to give him 35 in a row, breaking Matt Sladky's school record.
Harris made two more free throws in regulation but missed 2 of 4 in overtime and was 5-for-21 from the field in the game.