Arsenal showed their champion mentality on Saturday to keep coming at Hull despite initially struggling to create chances.
The Gunners went close through Declan Rice's first-half header, which was straight at Tzolakis, and Madueke's low strike from a tight angle which was saved after the restart, but were far from their best until their goal.
Mikel Arteta's team also had a penalty shout turned down in the first period after Kai Havertz's shirt was pulled in the box by Mendy.
The decision was not given on the field although the Video Assistant Referee told referee Farai Hallam to look at the pitchside monitor.
Hallam then informed the crowd Havertz had been fouled, but no penalty would be awarded as the forward had been in an offside position.
Similar to an incident in their win over Liverpool last week in which Jurrien Timber pulled Alexander Isak's shirt but the Reds were denied a penalty, the decision may have been headline news if Odegaard had not gone on to net the winner.
Instead Arsenal celebrated their ninth win in 11 matches and extended their unbeaten home run in all competitions to 11 games.