Other than moments of positive play and hard work Northern Ireland failed to yield enough clearcut opportunities. For Michael O'Neill, who has not won in nine matches at the helm, it was a bitter pill to take.
Israel looked like they would grab the game by the scruff of the neck when Swansea striker Itay Shechter struck the outside of the post from a short-corner routine, but thereafter the hosts had much the better of things.
Shane Ferguson's enthusiasm on the left was the catalyst, a neat flick and a testing cross lifting his side. Niall McGinn was the first to take his lead, beating Rami Gershon on the right before slightly over-hitting his cross.
A lengthy spell of possession ended with Paterson outmuscled by Tal Ben Haim as he looked to latch onto Davis' through ball in the 18th minute.
Northern Ireland's industry almost earned them a goal just after the half-hour mark when Paterson battled to win possession on the byline and cut back to Davis.
The skipper looked sure to convert from six yards but goalkeeper Dudu Aouate saw a powerful shot hit him flush in the face before Ferguson's reaction header was swept from the line by Yuval Shpungin.
The second half started slowly, Northern Ireland doing most of the pressing but without creating anything concrete.
Davis went close to a stunning opener in the 59th minute after McGinn's pacy counter-attack drove his side forward. The Southampton midfielder took down a loose ball with aplomb and then sent a 20-yard rocket inches wide of the top corner.
Paterson twice appealed penalties unsuccessfully, Ben Haim and Aouate both cleared by the referee after seeing the striker go to ground.
Then, in the 77th minute, Israel defied the run of play to take the lead. Israel substitute Refaelov was the scorer with a delightful finish across Carroll, having been teed up by a smart one-touch pass from Shechter.
O'Neill sent on Pat McCourt and David Healy in a bid to change his side's fortune but instead they leaked a second six minutes from time. The otherwise excellent Evans botched a defensive header, Shechter broke and Ben Basat finished clinically.