Yannick Sagbo won the first-half spot kick before being sent off after 26 minutes and Chris Hughton's Canaries were unable to take advantage of their numerical superiority
Steve Bruce's side saw the win through despite being reduced to 10 men midway through the first half when Yannick Sagbo was sent off for a headbutt.
The visitors had the better of the opening 10 minutes as Leroy Fer headed Steven Whittaker's cross over before Nathan Redmond stung the palms of Allan McGregor with a fierce drive from distance.
Bruce's side took the lead from the penalty spot on 22 minutes. Former Hull defender Michael Turner wrestled Sagbo to the floor in the six-yard box as he tried to reach Ahmed Elmohamady's cross from the right. From the spot, Brady stepped up to send John Ruddy the wrong way and break the deadlock.
Sagbo's eventful afternoon was soon finished, however, as he was dismissed by referee Mike Jones for an off-the-ball clash with Jonny Howson. The Ivorian appeared to headbutt the Norwich midfielder.
As the hosts focused on getting to halftime with the lead, chances were at a premium. Sone Aluko curled an effort beyond Ruddy's left-hand post, while Redmond fizzed a 25-yard free kick just wide of goal with the last action of the half.
Howson had the first effort of note in the second half, pinching the ball from Huddlestone before his deflected drive was parried down by McGregor.
Five minutes later, the midfielder took aim once more but this time his effort from distance flew over the crossbar.
Norwich continued to press. Redmond fed Ricky van Wolfswinkel, whose neat lay-off was lashed high and wide by Tettey from the edge of the box as Norwich became frustrated.
The visitors were brilliantly denied by McGregor with 20 minutes to go, the Scotland international leaping to tip Van Wolfswinkel's powerful header over the bar.
Norwich continued to dominate possession in the final stages but failed to create any further chances as Hull doggedly held on for its first points of the season.