Cosby scored five points in overtime for the Hilltoppers (9-7, 1-2 Conference USA), who won at home to avoid their first 0-3 start in conference play since 2010.
Florida Atlantic (2-14, 0-3) led by four with 2:01 left after Jackson Trapp made a pair of free throws, but Johnson hit a jumper and made three free throws to force the extra period.
Johnson made 7 of 9 from the field and was 2 of 2 in overtime.
The Owls have lost nine in a row and are 2-26 in their last 28 conference games.
Adonis Filer led FAU with 14 points, and Ronald Delph and Trapp each added 13.