American, leading 54-52, scored five straight points, four from Jesse Reed, to begin an 11-6 run in the final minute.
Jones was 9 of 14 from the field and hit three 3-pointers. Jalen Rhea scored his 15 points on five 3-pointers and Reed added 12, going 6 of 6 from the free-throw line. The Eagles were 9 of 23 from 3-point range.
American (4-15, 2-6 Patriot League) has won two in a row after losing 10 straight.
American pulled off the upset despite coming in as the nation's second-to-last team in scoring offense (54.5) while Navy came in ninth in scoring defense (61.7).