Omar Bogle came close twice in the first half, forcing Yeovil goalkeeper Artur Krysiak to palm a stinging drive over the bar before shooting narrowly wide from the edge of the box, while James Berrett had a good chance to win the game late on when his long-range effort was well stopped by the Polish keeper.
It was a performance that pleased caretaker boss Dave Moore, who admitted that he would have been satisfied with a stalemate against a Glovers side that had won their last three matches before kick-off.
Moore said: "If you'd have said to me yesterday morning, 'I'll give you a point', would I have got on the coach? Probably not.
"I'd have taken it, but then you come away and you think actually we might have done a bit better, but overall I'm satisfied with the point."
"I thought we were the better team in the first half, I thought they were perhaps a bit better certainly for a period in the second half but then I thought we got back into it and might have pinched something at the end.
"We certainly had some good spells and composed possession of the ball and looked threatening in the first half."
Grimsby had an appeal for a penalty in the fourth minute when Bogle went down under the challenge of Darren Ward, but while Moore disagreed with referee Brendan Malone's decision not to award a spot-kick, he stopped short of criticising the official.
He added: "I thought it was a penalty. Omar thought it was a penalty.
"He said 'why would I go down when I was going to get my shot off?' but that's the referee's decision and I'm not moaning about it.
"I just thought it was and it would've been nice but there you go, that's football."
Despite seeing his side's three-game winning streak come to an end, Yeovil boss Darren Way described the result as his best since he took the helm at Huish Park last December.
The game extended the Glovers' run of clean sheets to four matches and continues a streak that has seen them climb from the relegation zone to the top half in the space of six weeks.
Way said: "That for me is my biggest result since I've been a manager, knowing what we were up against today.
"I have to give them credit, they're a very good team, every time they received the ball they've got pace, they've got power, they can cause teams a lot of threats and it was the way we stood strong and were very resilient, that's what you need within your team to be successful.
"I think the players are very proud of that achievement [keeping four straight clean sheets].
"They had nine last year so it shows the system is working, I think they're working very hard at that system but as I said it's just that end product in the final third because the hardest thing to do in a football match is score goals."