Ryan Lowe's 41st-minute penalty was the deciding factor in a dour game that had moved City out of the relegation zone, until Grimsby overhauled them following their 2-0 win at Notts County.
"We won a nasty game," said Wright. "There wasn't loads of football played and the lads were nervy after Saturday's defeat but they put in a battling performance.
"It's weird as when you are on the sidelines you are more nervous than when you are playing. I have played in front of 100,000 and have not been nervous but on the bench you are kicking every ball and putting in every tackle.
"The main thing at this stage of the season is to get three points, it doesn't matter about the performance. There is a lot of pressure on us and we didn't do well against Macclesfield on Saturday.
"But we showed a lot of character and dug in and the win is all that counts in a nasty game."