Trailing 28-24, Alexander, who rushed for 50 yards, scored on a 15-yard run with 4:29 left in the game. The Cowboys missed the extra point.
Florida Atlantic had a final possession, but on fourth-and-3 at the Wyoming 47-yard line, Rusty Smith's pass was batted down by Mike Neuhaus to seal the win.
Austyn Carta-Samuels was 29-for-38 for 223 yards for Wyoming (3-2). Brandon Stewart also scored a touchdown for the Cowboys.
Rusty Smith was 17-for-28 for 235 yards for Florida Atlantic (0-4) and tossed a touchdown pass to Darian Williams. Chris Bonner and Alfred Morris had rushing touchdowns for the Owls.
Alexander's 3-yard run in the third quarter gave Wyoming a 24-14 lead.
Morris' 1-yard touchdown run for Florida Atlantic closed the gap to 24-21.