Chris Holland and Darren Edwards scored either side of half-time to fire Blue Square South minnows Bath into the second round.
Managerless Grimsby had no answer to the visitors' attack-minded approach.
And boss Britton beamed: "It would've been a travesty had the result been any different as we were clearly the better team.
"It's a fantastic result and we're absolutely elated - this means so much to everyone at this club.
"Not many people would have given us a chance coming here but we believed in ourselves and no-one can argue with this result.
"And more importantly for me as the manager it was the manner in which we did it. We took the game to Grimsby and deserved it and the performance was excellent from start to finish."
After a scrappy start, the non-leaguers settled and nearly created a chance on five minutes when Kaid Mohamed outpaced Bradley Wood but could not find Lewis Hogg with a square pass.
Grimsby's best chance came four minutes later but Barry Conlon headed over from close range.
It was a rare bright spot for the hosts and they went behind on 32 minutes when Holland headed home Hogg's superb free-kick from the right. And Grimsby's task was made harder when Bath striker Edwards rifled a deflected shot into the top corner.
And despite Grimsby's superior fitness in the final quarter, Bath held out and could have even increased the scoreline had Mohamed and Hogg not seen shots saved by keeper Nick Colgan.
It was a disappointing exit for woeful Grimsby, their fifth first-round defeat in six years.
And caretaker-boss Neil Woods had no complaints.
He said: "We weren't good enough and there are no excuses from me. Credit to Bath they deserved to win. We had a good chance early on for Barry Conlon but didn't create anything else after that.
"Too many players didn't take any responsibility and for whatever reason we were just not good enough."
Goal hero Holland, a PE teacher, said: "We matched Grimsby in every area and weren't going to lay down although their fitness told towards the end,
"We knew we had to start well and hold them off and we grew into the game to get a good lead.
"It was a great delivery by Lewis, one of those crosses just waiting for someone to attack and I managed to get enough on it to see it go into the back of the net.
"The main concern this season has been the lack of goals really but we got two today and could have had more in the second half.
"We deserved to win and the fans were brilliant. It was a long trip but they are happy."
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