Mikel Arteta's side put on a show for their travelling supporters as they recorded their third consecutive victory to go top of the league after beating Crystal Palace and Leicester City -- the first time they have opened a Premier League season with three wins in 18 years.
"Still far [from perfect]," Arteta said in an interview with Sky Sports.
"We had some strong periods. In the second half we conceded too many chances for Bournemouth. I'm proud, it's not easy to come here and win 3-0 but there are still things to prove.
"We started the game really well. We were dominant, precise. The two goals gave us a lot of confidence. Jesus was involved in the goals. His contribution to the team was outstanding."
Arteta said there were areas of the squad which still needed "more firepower" before the end of the transfer window, although he said any signings would not be made for the sake of it.
"Obviously we want to get the players that we believe will make a big impact in the team," Arteta said.
"If we can do that, we will try to do that. If we cannot we will stay with the players we have.
"It doesn't change, but we know as well where we are short, where we have to add more firepower. And if we can do it, we will do it."
Arsenal made the perfect start in the fifth minute when Jesus waltzed through the Bournemouth defence and offloaded to Gabriel Martinelli. The winger's low shot was saved by goalkeeper Mark Travers, but Odegaard was on hand to slot home.
Odegaard grabbed his second goal just under six minutes later after a low cross from Ben White, taking advantage of more intelligent movement from Jesus to fire past Travers and put Arsenal in full control.
Arteta's side rarely looked troubled in the first half and their fans were in buoyant mood as the team clicked through the gears with little resistance from newly promoted Bournemouth.
The hosts emerged with renewed purpose after the break, but that positivity disappeared when William Saliba produced a fine finish in the 54th minute, sweeping home a cut-back from Martinelli with his left boot to score his first goal for the club.
Jesus looked to have put further gloss on the victory in the 72nd minute when he latched on to a clever pass from Odegaard before dinking the ball over Travers, but it was ruled offside after a VAR review.
"He is not just about goals," Odegaard said of Jesus. "He gives us so much in the play and he drops down to combine.
"He is brilliant and he is amazing to play with. We are lucky to have him here."
Bournemouth had an opportunity to cut the deficit in the 79th minute when Kieffer Moore rose to meet a cross, but goalkeeper Aaron Ramsdale tipped away the striker's looping header in acrobatic fashion.
Odegaard added: "Compared to last season we are a little bit ahead but we have to stay calm and we have to stay humble and see where it takes us."