Dyson's shot put the Hawks ahead 65-62. Justin Graham, who finished with 12 points off the bench, made two free throws with 25 seconds left and one of two at six seconds that proved to be the difference when Jaleen Smith hit a 3-pointer for the Wildcats with two seconds left.
Mark Nwakamma also had 16 points for the Hawks (9-7, 2-1 American East).
Hartford had a 10-point lead in the first half but New Hampshire rallied to go up 31-29 at the break. The Hawks twice led by six in the second half.
But Hartford made 20 of 25 free throws while New Hampshire went 13 of 18.