The result has left both teams with 2-2 records in EuroChallenge Group E.
Omar Sneed scored 27 points for Hapoel.
Trailing 59-48 in the final minute of the third quarter, the Israelis turned the game around in the fourth quarter when they outscored the Dutch hosts 24-14.
Sneed's three-point play with 2:12 remaining put them in front for the first time in the fourth quarter at 68-66.
The home side went back into a 69-68 advantage, however, on a three-point play from Clint Harrison.
The game was hanging in the balance with 35 seconds left when Bowers drew a foul and made two free throws to tie the game at 72-72.
With 15 seconds to play, EclipseJet MyGuide got the ball inside to Avis Wyatt and he drew a foul.
Wyatt made just one of two free throws, leaving the door open for Bowers who made his two free throws at the end for victory.
Teddy Gipson poured in 22 points for EclipseJet MyGuide.
Arik Shivek, EclipseJet MG Amsterdam coach:
"We didn't lose the game when Avis Wyatt or Teddy Gipson missed the shots. We lost when we played alley hoops instead of a simple lay up in two-against-one situations, when we were ten points ahead."
Stefan Wessels, EclipseJet MG Amsterdam:
"It is a sad loss for us, but we have to move on."
Guy Goodes, Hapoel Jerusalem coach:
"The tables were turned this time. In Jerusalem we didn't make this shot and this time we made it."
Timothy Bowers, Hapoel Jerusalem player:
"I'm happy with the win. We worked hard for the victory."