A-League Men champions City are in a difficult patch of form in the league, which they are juggling with an impressive run in the Asian Champions League Elite competition.
City were seeking a response to their 4-1 loss to Adelaide, while the Mariners needed to steady after their 3-2 defeat to Western Sydney.
In a largely dour affair, both teams had goals chalked off for offside.
City had 14 shots to Central Coast's four, with Redmayne making five saves.
The 36-year-old former Sydney FC goalkeeper and capped Socceroo has found a new lease of life since returning to Central Coast.
"Coming up against Melbourne City, I think as a goalkeeper you're always going to be a bit busy," Redmayne told Paramount Plus.
"I'm not sure what to make of that game. A point's probably fair in the end.
"I thought we could have snatched three points at the end there, but a point's probably a fair result.
"(I'm) loving my football. I enjoy playing with a smile on my face, and there's a few boys in this team that enjoy doing the same thing ... really enjoying Coast life."
The gritty draw came at a cost, with Mariners skipper Nathan Paull limping off in the 71st minute.
Takeshi Kanamori forced a good save from Redmayne in the 24th minute.
Redmayne then did brilliantly to snap up a bobbling strike from Kanamori in the 37th minute
But otherwise Central Coast had lots of promising moments, with Lucas Mauragis particularly dynamic down the left, while Miguel Di Pizio had a penalty shout waved away in the 19th minute.
The Mariners thought they they had snatched a deserved opening goal in the 42nd minute when Sabit Ngor tapped home from Mauragis's excellent run and squaring ball.
But the striker hadn't checked his run and was offside.
Both teams kept pressing for a winner, and City goalkeeper Patrick Beach made a strong near-post save to deny Nicholas Duarte in the 86th minute.
City striker Andrew Nabbout thought he'd scored the winner in the 90th minute after tapping home from a bizarre deflection off a teammate, but he was offside.
The eighth-placed Mariners (2-2-2) next host Sydney FC on Saturday, while fifth-placed City (2-3-1) travel to play Newcastle.
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