Granit Xhaka and Pierre-Emerick Aubameyang struck as Arsenal recovered from an early Luka Milivojevic penalty to take the lead at Selhurst Park, but the Serbian midfielder converted again in the closing stages to share the spoils.
Ozil was substituted after 68 minutes having made just one key pass for his side, and Souness joked that he showed more emotion as walked off than he ever has on the field before.
"To be fair, Ozil showed a great deal of passion when he was subbed today," he said on Sky Sports. "That's the most I've ever seen him show in a game of football.
"You're the captain, when you're on the pitch show courage to get on the ball when you're not dominating. Show real passion, maybe you're not built for it but certainly make an attempt to make a few challenges and win the ball back for your team.
"He doesn't do that. I'm sorry to harp on about him a bit but a man with that much talent should influence games like that."
Souness has called out the attacking midfielder already this season, prompting Ozil to say this week that he merely laughs off the former Liverpool star and manager's critiques.
"I don't listen to these people - I just listen to my coach and my people who say things straight to my face," he said. "So of course, if I play bad, I know that just as I know when I play well.
"The most important thing is that the coach can help me - if I make mistakes he can tell me, we can speak and I can improve for the future.
"But if people just say bad things about me because they want to be in the newspaper - I know that and I don't care about them."