Al Qadsia were beaten 2-0 by domestic rivals Al Kuwait in last season's final, but they gave themselves a great chance of avenging that defeat with a first-leg victory at the Al-Sadaqua Walsalam Stadium.
Persipura had scored 17 goals during the knockout stages before the last-four contest, and with two away goals to take with them back to Indonesia they will retain hope of overturning the deficit in the second leg.
Subotic opened the scoring in the 12th minute, heading home when unmarked in the area.
However, Persipura levelled nine minutes late with a superb long-range effort from Titus Bonai.
The visitors were then in front shortly after the half-hour mark, when Talal Al Amer put the ball into his own net.
That lead lasted for over 30 minutes, but Al Qadsi mounted a fightback of their own and their recovery began in the 64th minute with a leveller from Saif Al Hashan.
Al Qadsia then moved ahead for the second in time the match just two minutes later, when Bader Al Mutawa finished with a fine volley.
Subotic then strengthened Al Qadsia's grip on the tie with a well-placed finish into the bottom left-hand corner, ensuring they will start the second leg in a fortnight with a two-goal lead.