Fouch was 0 for 6 from the field in overtime until he hit a three-pointer to tie it at 62 with 19 seconds left.
One second later, however, Fouch fouled Randal Holt, who made two free throws.
"He just grabbed him," Flint said of Fouch's foul. "It happens."
Fouch thought he had hit a two-point basket and the Dragons were trailing and was more critical of himself.
"Disappointing, that's all," Fouch said. "I've got to do better."
Holt hit two more foul shots with three seconds left and later called the win a "huge one" for the Golden Flashes, who went 21-12 last season to make the postseason for the 13th time in 14 years.
"Playing Temple next week [Tuesday], we sure didn't want to be 0-1," Holt said.
Fouch kept the Dragons in the game with his three-pointers. The senior sixth man shot 6 of 12 from beyond the arc and made all six of his free throws, but missed all seven of his two-point shots.
Chris Evans turned on the baseline and hit a short jumper for Kent State to tie it at 55 with 26.9 seconds to play in regulation.
Dragons point guard Frantz Massenat then dribbled the ball out top before deciding to drive - only to turn it over without making a pass or getting off a shot with 4.3 seconds left.
Evans led Kent State with 21 points and Holt had 17. Damion Lee and Kazembe Abif, who fouled out in overtime, each scored eight points for Drexel, which shot just 37 percent from the field.
"We didn't come ready to play and they did," Fouch said. "I hope that's a lesson. We had trouble with it last year and we can't keep doing it."