Al Jaish needed to win by two goals to advance and opened the scoring through Maher Al Said 10 minutes before the break.
But despite Siarhei Krot late’s dismissal, Filiposyan headed home to end Al Jaish’s hopes as Nasaf joined Kuwait’s Kazma in the last 16.
Needing to win Al Jaish made a quick start as midfielder Yasr Akra fired over after six minutes.
The home side continued to dominate and captain Maher Al Said was next to try his luck, but again failed to find the target as his effort flew well wide of the upright.
But after 35 minutes a swift counterattack pushed Al Jaish forward and Al Said by-passed defender Artyom Filiposyan before firing past Nasaf goalkeeper Dmytro Kozachenko.
Nasaf nearly replied three minuets before the break but substitute midfielder Jahongir Jiyamuradov was denied by goalkeeper Taha Mosa.
The visitors continued to improve into the second half and came close to equalising four minutes after the break when Botir Qodirqulov’s well-stuck effort sailed over the crossbar.
Six minutes later Kozachenko was called into action to beat away a strike from Abdulrazak Al Husein.
Syrian midfielder Al Husein was again denied six minutes after the hour mark when Kozachenko parried his strike from outside the area before Akra failed to find the target following good work from Al Said.
Nasaf came close with 14 minutes remaining when substitute midfielder Rustam Kadirov beat Al Jaish goalkeeper Taha Mosa with a powerful close range shot but Feras Ismail was on hand to clear off the line.
And Al Jaish’s cause was handed another substantial boost a minute later as Nasaf substitute Krot received a straight red card after appearing to lash out at an opponent.
Al Jaish should have taken full advantage with eight minutes remaining but Kozachenko came to Nasaf’s rescue to deny Al Husein.
Al Husein shot over three minutes later, but a late spell of Nasaf pressure resulted in Filiposyan heading past Mosa to deny the home side.