The Norwegian was named man of the match in the Community Shield win at the Principality Stadium in Cardiff, capping a dominant Arsenal display with a goal to add to Riccardo Calafiori's early opener and a Kai Havertz strike.
The result has left Ødegaard hungry for more as Arsenal prepare to open their Premier League title defence against Coventry City on Friday.
"We wanted to show our level," he said. "Of course, last year we won the league. But this year we want more.
"There is a lot more to play for. We want to do it again in the league and then there are the other trophies. There is a lot to work for and still a lot of room for improvement."
Ødegaard reserved particular praise for £75 million signing Bruno Guimarães, who made his first Arsenal start since his big-money move from Newcastle before being replaced with an ice pack at half-time. The Brazilian settled quickly into the side, driving forward with intent and looking for attacking passes throughout his 45 minutes on the pitch.
Ødegaard, who has previously clashed with Guimarães during their days facing off in Premier League midfield battles, said the pair had found an instant understanding in Arsenal colours.
"It was really good, really natural. Since the first day he came in, it has been such a natural thing. Although we had some big fights with him in the past, he has just come into the team as a real natural part of the team."
He went on to detail exactly what makes his new midfield partner stand out.
"And you can see his quality. He can win the ball, he's a fighter, but he also has so much quality on the ball. He has that, what can I say… he is so calm on the ball. He always takes the right amount of time and he finds the pass."
The captain added that the impression had been immediate from Guimarães' very first involvement in training.
"From the first second he is in training you can feel that presence and quality. He is a brilliant player. I am looking forward to it."
Arsenal's win over City marked the club's 18th Community Shield title and continued a perfect run in their last six appearances in the fixture, offering an early marker of intent before the serious business of the Premier League season gets under way this weekend.
Ødegaard himself ended a personal drought of 20 matches without a goal, netting for the first time since December, in what was as much a personal statement as a collective one for the Arsenal captain.