Vailas, who earned Colonial Athletic Association Conference Offensive Player of the Week in the 52-19 win over Dartmouth last Saturday, completed 17 of 24 attempts with no interceptions. Vailas did not play in the second half as the Wildcats had a 34-0 lead after two quarters.
New Hampshire (4-1, 2-0 CAA) has now won four consecutive games.
Elon (1-4, 0-1) got its initial first down as the first quarter ended when the Phoenix drove to New Hampshire's 44. UNH's Steven Thames stepped in front of a Mike Quinn pass and returned it 55 yards for a 24-0 lead to start the second quarter.
R.J. Harris, who also only played in the first half, finished with 77 yards on five carries.