Peacock's 38th-minute goal was enough to seal all three points for Grimsby, for whom the biggest plus - other than the result - was an impressive backline.
Summer signings Lee Ridley and Steven Watt stood out against Crawley, who had dominated spending in the Blue Square Bet Premier this summer, had most of the possession despite playing almost an hour with 10 men.
"It was a very good defensive performance, we defended well all the way through like we didn't last year. On the ball we were way off and know we're a lot better than that, but we're very happy with the result," he told the BBC.
"We just ground out a very hard win. Even when they were down to 10 men they were passing it round like they had one more player than us. It's a weird thing, sometimes playing against 10 men can go against you.
"We never made the ball do the work for us, we didn't pass the ball anywhere near the level we should have. That's something we've got to learn to do. It was just an off-day, that's all it was."
Peacock's free-kick was unusual, rather than blasting the ball he simply side-footed it around a defensive wall that had been badly lined-up.
"There was a massive gap, the wall didn't seem lined up to great and I could just pass it in," he added.
"We've been working on set-pieces all week and I think it showed, we were a threat from them."