Odegaard scored the opening goal in Arsenal’s 3-1 win over Olympiacos in the Europa League on Thursday evening.
Smith Rowe, meanwhile, came on as a late substitute after missing the 1-1 draw with Burnley last Saturday due to injury.
Odegaard and Smith Rowe are competing for the same spot in Mikel Arteta’s side ahead of the north London derby this weekend.
And Keown believes Odegaard is displaying the ‘wrong attitude’ after the on-loan Real Madrid midfielder voiced his fears about making mistakes
‘I think he’s a really interesting player for Arteta,’ Keown told BT Sport.
‘When I listen to him it’s what I see on the pitch. He talks about the team a lot, doesn’t really talk about himself.
‘The spotlight has been on him since he was a 16-year-old kid who went to Real Madrid. Maybe he should be a little bit more willing to talk about himself.
‘It’s about him now. This goal should elevate him now. Lift him. Give him the confidence he needs to express himself.
‘He talks about being frightened of making mistakes. That’s the wrong attitude.
‘You’ve got to love it. You’ve got to play with freedom. Be free in your mind and Wenger used to say, ‘just relax and play, play what’s on’.
‘And I think there’s something there under the bonnet. We saw a glimpse of that with that strike. That is a top drawer effort he’s hit there.
‘But there are other mistakes in the game and just needs confidence. I think he’s been in a difficult situation, it’s an interesting signing.
‘I’m not even sure he’s going to play at the weekend because Smith Rowe is fit.’
‘He knows where to be defensively when they press and he drops back into midfield when they’re allowing the teams out in that secondary position a little bit better [than Odegaard].
‘He plays the ball forwards and he runs and moves, but that’s because he’s probably ahead confidence wise. Still, Odegaard can offer something for Arsenal this season.’