Hapoel Tel Aviv must overturn a one-goal deficit in Israel next Tuesday if they are to keep alive their hopes of reaching the group stages of the Champions League.
A second half penalty from Aktobe skipper Samat Smakov gave the hosts a narrow advantage from Wednesdays third qualifying first leg in Kazakhstan.
The Israeli champions featured keeper Vincent Enyeama, who represented Nigeria at last month's World Cup, while Ben Sahar, the former Chelsea striker who recently joined on a 12-month loan from Espanyol, came on in the second half.
Aktobe had the better of the match with 18 shots compared to Hapoel's six.
Hapoel: Enyeama, da Silva, Shechter, Badir (capt), Vermut, Zahavi, Lala (sub: Sahar 57), Ben Dayan, Kanada, Yadin, Rocchi (sub: Shivhon 66)
Booked: Lala 41, da Silva 66