The Volunteers, seeded second in the East Region, struggled for much of the game to shake the underdog Eagles.
Tennessee led by just 53-51 with 5:45 left but put the game away with a 19-6 run.
Garrison Carr's 26 points led American.
Brian Gilmore added 13 points and six rebounds for the 15th-seeded Eagles, who ended the season with a 21-12 mark.
American, which won the Patriot League Tournament, was playing in its first NCAA Tournament game.