Davis and Terry Allen finished with 11 points apiece.
Isaiah Miles pulled Saint Joseph's (12-16, 6-10) within three and Aaron Brown added a free throw to make it 59-57. Anthony made two free throws to put Richmond back up by four and sealed it with a layup.
Brown had 15 points to lead the Hawks, who went on a 6-0 run to tie the game at 54 but never led after halftime.
Richmond opened the second half with a 12-4 run to lead by 14. The Hawks answered with a 15-5 run to get within 48-44 with 8:47 left.