In a pulsating start to start the NRL season on Thursday, Harry Grant darted over from dummy-half in the fourth minute of extra time to keep Melbourne's incredible round-one record under Bellamy alive.
It came on a drama-filled night, with Munster requiring two pain-killing injections in his finger after a first-half injury.
He will likely require surgery, while Xavier Coates is also set for a stint on the sideline with a collarbone problem.
Already decimated by injuries in the pre-season, Melbourne were forced to fight back from behind twice as they maintained their perfect record in round-one matches dating back to 2002.
But for several stages in the opener, that run looked over.
Melbourne trailed 6-0 at halftime and went behind 12-6 midway through the second, as the Eels enjoyed the majority of early ball in the match.
Then when they needed him most, it was Munster who stood up.
After making a defensive lapse in the lead up to Parramatta's go-ahead try, the injured five-eighth dummied and skipped his way across field to put Young Tonumaipea over.
Nick Meaney iced the conversion with 16 minutes to play, creating the 12-12 deadlock that would remain until the end of regulation time.
Melbourne's win comes with Ryan Papenhuyzen unlikely to play in the opening eight rounds of the season, as he battles to recover from his fractured knee cap.
Justin Olam, Tariq Sims, Tui Kamikamica and George Jennings are among other injured troops.
"Just being so gutsy through the whole 80 minutes, I don't think we have been like that as a team for the past 12 months," coach Bellamy said.
"I thought we made some steps forward in that area and hopefully we can keep it up.
"If we keep showing the fight they did tonight, I am sure we'll be okay."
Parramatta were also missing several of their grand-final stars, but will be left to rue not being able to ice the match.
They were on top when Clint Gutherson put Will Penisni over for the first try of the season when the Eels skipper busted through a tackle and found his centre unmarked.
With scores level at 6-6, it looked as if Josh Hodgson would be the hero for the Eels in the first game for his new club and off the back of an ACL injury at the start of 2022.
Playing his first game in a year, Hodgson spotted Munster sprinting out of the line in defence and found Junior Paulo to barge his way over to set up the 12-6 lead.
But from there, Parramatta blew their chances.
Maika Sivo kicked a ball dead on the first tackle when the Eels had a scrum in the Storm's half, while halfback Mitch Moses had a field-goal attempt charged down with four minutes to play.
"I felt like we did enough to win it, but we just didn't execute enough," coach Brad Arthur said.
"It's not like a heap of errors, but there were some big moments at the end that we just didn't get right.
"It's not that (Sivo) one in particular, but six or seven minutes to go from a scrum, do we need to try and score?
"We probably need to build a bit of pressure there, keep it simple and set for a field goal. It doesn't matter if win 18-12 or 13-12 with a field goal."
