The Gers boss had presided over three victories and one draw since replacing Paul Le Guen at the Ibrox helm last month.
And Smith's compact side had looked on course to secure a 1-1 draw ahead of next week's second leg when Nacho Novo bundled home a second-half equaliser in Tel Aviv.
But a 75th minute winner from Baruch Dego has left Rangers' hopes of progressing to the latter stages of the competition hanging in the balance.
In a scrappy opening 30 minutes, Rangers were almost gifted the lead when goalkeeper Shavit Elimelech clattered the ball against Novo but the lively striker couldn't take advantage.
Novo then flicked a header from a Chris Burke centre across the face of goal and narrowly wide of the far post as the visitors continued to probe.
But despite having the best chances of the first period, it was the Israeli side who drew first blood through Salim Toema.
David Weir was adjudged to have fouled Ibe Ogbona just outside the box and Toema dispatched a superb 25-yard free-kick with his left foot over the wall and into the top corner of the net to silence the 1,000-strong travelling contingent.
Rangers though refused to be downhearted and Novo pulled them level eight minutes after the half-time interval.
Barry Ferguson's 20-yard free-kick took a deflection and struck the post before Novo forced it over the line with his knee.
Hapoel bounced back and came close to reclaiming their lead on 70 minutes when Elyaniv Barda met Toema's corner but hurried a shot and crashed his effort over the bar from close range.
But Rangers failed to heed the warning and Dego fired the home side into a 75th minute lead.
Barda mis-hit a cross from the right side of the box but the bell fell to Dego who fired home from inside the six-yard box to settle the match.