Doncaster manager Darren Moore bemoaned a lack of cutting edge for his side’s exit from the Carabao Cup.
League One Rovers enjoyed the bulk of possession and also forged the better chances at Grimsby but lost to a 39th-minute goal from Jordan Cook.
Moore said afterwards: “It’s a game that we dominated from start to finish. We were totally dominant in all areas of the pitch, but it was just in that final third.
“It was just that finishing touch, but we created enough chances.
“We had enough chances to win two games, really, but it’s about putting the ball in the back of the net.
“The performance was there, the work-rate was there and the endeavour was there.
“The one word I would say to the lads is to be ruthless, because we created enough chances.”
Another home draw awaits for League Two Grimsby, with the Mariners drawn against Macclesfield in the second round.
Cook, making his first appearance of the season, got the only goal of the game with a composed finish.
However, before going behind to that solitary Cook strike, Ben Whiteman, Kieran Sadlier and Niall Ennis all passed up opportunities for the visitors.
Grimsby boss Michael Jolley said: “Doncaster are a good side and they play some good football. They created chances in the first half, but we are really pleased with the outcome.
“We made a small tactical change in the first half and we feel that it helped us – and the goal then gave us more confidence.
“Credit to the players. It was a really good goal. We do a lot of work on counter-attacking. It was a good break from Jordan and a good pass to Moses (Ogbu).
“The weight and timing of his pass was excellent and it was a great finish.”