The visitors to Nicosia were beginning their campaign after their fixture with Belgium was suspended and eventually secured the lead which their dominance warranted.
It was Cyprus who started the brighter as Andreas Makris and Konstantinos Makridis tried their luck from range, but Israel soon took control of possession and eventually a series of chances came their way.
Ben Haim, namesake of the former Chelsea and Manchester City defender who was also in the Israel lineup, tested Cyprus goalkeeper Antonis Georgallides with a 20-yard drive minutes before Damari's opener.
The striker found the bottom corner from Ben Haim's cross on 38 minutes, and Eran Zahavi had a good opportunity to double Israel's lead moments later when he struck straight at Georgallides from close range.
Ben Haim did put the visitors 2-0 up prior to the interval as he impressively weaved his way through several challenges in the penalty area before finishing smartly to leave Cyprus, who beat Bosnia-Herzegovina last time out, with a mountain to climb upon the restart.