The Kumuls skipper scored a try and kicked four goals in front of 6273 fans at the Halliwell Jones Stadium in Warrington as PNG bounced back from last week's narrow loss to Tonga.
In a standout performance, Martin's first conversion attempt denied him a place in the record books with 42 back-to-back successful goals when he failed to convert a Rodrick Tai try in the 14th minute.
Tai was a constant threat for the Kumuls, scoring two tries and clocking 164 running metres.
The victory means PNG must beat winless Wales to set up a likely quarter-final clash with hosts England.
Centre Justin Olam was taken off in the second half after scoring a late try when the result was assured.
The Cook Islands, meanwhile, face an uphill battle to qualify for the last eight, with group D heavyweights Tonga to come in their final pool match.
Tai grabbed the opener, running 80 metres from a 20-metre tap to score in the corner after Cook Islands winger Steven Marsters had kicked a penalty.
Martin was unable to convert Tai's effort but made amends when five-eighth Kyle Laybutt crashed over soon after.
Paul Ulberg hit back for the Kukis to cut PNG's lead to 10-8 at halftime after Kumuls forward MacKenzie Yei was sinbinned for a high shot.
The Kumuls pulled ahead after the break, with Martin, Lachlan Lam, Tai and Olam all going over.
Cook Islands centre Anthony Gelling and fullback Kayal Iro scored late but their hopes of a comeback were foiled when Reubenn Rennie was sinbinned by referee Chris Kendall for a tip-tackle on PNG prop Wellington Albert.
