The 21-year-old midfielder, who was making his first start for Rangers, fired low into the net from 12 yards after a Robbie Manson effort had been blocked.
Greenhill had also come close in the first half but his powerful shot from just inside the penalty box was tipped over the bar by Mark Peat.
Chances were few and far between as both sides struggled to create any fluent moves on the sun-baked pitch.
Arbroath's best opportunity fell to Alan Brazil a few minutes before the interval but the striker, who is on the comeback trail after six months out with a broken leg, weekly hit the ball straight at Gary O'Connor.
Berwick's Steven Noble shot into the side-netting at the start of the second half and Gary Wood came close with a curling shot which went narrowly over the bar.
Arbroath pressed for an equaliser but the visitor's defence held firm.
Berwick assistant-manager Matt Carr felt his side deserved the victory. "It's a great result for us," he said. "Not many teams will come here and win.
"We are rebuilding a team here and I think we have the makings of a fine side."