Sedgemore scored the only goal of the game with a free-kick mid-way through the first half to ensure Shrewsbury went their third game in a row unbeaten.
Town boss Gary Peters was pleased with the win, but admitted it was a nervous second half. He told ShrewsWeb: "The second half was one of the longest 45-minutes of my life, when your 1-0 up and hanging on like that it gets you going a bit."
He also explained how Town's adapted tactics played their part, "They had some dangerous players, so we changed our strategy slightly to try and stop the ball getting into the front two. We put Jake Sedgemore in front of their front two to protect our centre backs," Peters added.
One downpoint from the victory was the injury to Darren Moss, who was stretchered off on Saturday. However, it looks as if the problem is just bruising and Moss was walking by the end of the day.
Grimsby boss Russell Slade was not happy with the result. He said: "I was massively disappointed with the result as were the players. I did not think that there was enough about us in the first half in terms of creating chances but you have to take the windy conditions into account."
He also paid tribute to the Town defending, saying: "They were resilient, defending in numbers, and they have come and done a job."
Town's next fixture is the home clash with relegation strugglers Kidderminster, with the Gay Meadow side looking to pull away from the drop zone.