The hosts struggled to break down Hull's disciplined defence for much of the encounter, despite creating several promising opportunities and feeling aggrieved after an early penalty appeal was turned down.
Arsenal came close to taking the lead through Declan Rice, whose effort from the edge of the penalty area went close as the Gunners searched for a breakthrough.
David Raya was also called into action, producing an important save to keep out Elliot Stroud's dangerous low delivery into the box as Hull threatened on the counterattack.
The visitors remained organised defensively, frustrating Arsenal and limiting their opportunities in the final third.
Kai Havertz had a difficult afternoon, missing an opportunity to make an impact before being involved in a penalty incident following a shirt pull inside the box. However, a VAR review ruled him offside, denying Arsenal the chance to take the lead from the spot.
With the game threatening to end in frustration for the hosts, Arsenal turned to their substitutes and increased the pressure in search of a decisive goal.
The breakthrough eventually arrived through Odegaard, who reacted quickest after Noni Madueke's effort was pushed back into a dangerous area by Hull goalkeeper Konstantinos Tzolakis.
The Arsenal captain was on hand to tap home the rebound, sending the Emirates Stadium into celebration and securing a crucial victory for the Gunners.
Madueke was also instrumental in the decisive moment, having provided one of Arsenal's biggest attacking threats throughout the match.
Odegaard's performance earned him the highest average rating from BBC Sport users, receiving 7.40 out of 10 to claim the Player of the Match award.
David Raya followed with an average rating of 7.17, while Madueke collected 7.12. Gabriel Magalhaes and Ezri Konsa also impressed, receiving ratings of 7.10 and 7.08 respectively.
Despite the defeat, Hull goalkeeper Tzolakis earned praise for his performance, although his parry of Madueke's effort ultimately led to the decisive goal.
The result gives Arsenal an important victory after a challenging afternoon against a Hull side that defended resolutely but were unable to hold on until the final whistle.
