The Kuwaiti outfit claimed a 4-2 home win in the first leg thanks to three goals in the final half-hour.
It sets up Antonio Puche Vicente's team ahead of their visit to Jayapura.
Danijel Subotic was the hero for Al Qadsia in the first leg, netting a brace as they took control against their Indonesian opponents.
Al Qadsia were runners-up last year and in 2010, making them favourites to reach the decider once more.
Persipura need goals and showed they know how to score, taking a 2-1 lead in the first half at Kuwait City.
Titus Bonai and Boas Solossa were on the scoresheet away from home and need to produce similar in the return leg.
Coach Jacksen Tiago will be hoping his attack can click – and his defence can keep a clean sheet – as Persipura look to reach their maiden AFC Cup final.
