The Gunners have been hit hard by injuries this season and Saka is the latest player to be struck down.
It comes off the back of Gunners captain Odegaard injuring his ankle in September while playing for Norway and not returning until the start of November.
Arteta has revealed he is now coming up with plans on how Arsenal cope without Saka ahead of a busy January schedule.
And asked if that, and the time Arsenal were without Odegaard, will make him a better coach, Arteta said: “I think so, yeah.
“And obviously we started the season with one of the thinnest squads in the Premier League, we knew that.
Bukayo Saka and Martin Odegaard have both suffered significant injuries this season
Arsenal FC via Getty Images
“Okay, so what do we need then? If we are the thinnest, what do we need? And really dig into that and make sure that we utilise every player in the best possible potential, everybody has to feel part of it.
“You need to rediscover players in different positions, relationships. It’s a great one, honestly. It’s a really good exercise and the boys are willing to do anything we demand them to do, so that’s a really good thing as well.”
Saka’s hamstring injury is the first major one of his career and he left Selhurst Park on Saturday night with the aid of crutches.
The winger underwent assessment over the weekend and on Monday, and is now set for a lengthy spell out.
“He was gutted,” said Arteta. “You can tell that he has not been injured because he was really, really emotional and really down.
“And we need to lift him up. It is a big part of his job. And he will be fine but he is going to need a few days.
“He will immediately realise that he is so important in that dressing room. His energy level, his body language, and how he communicates with the rest of the team has to be really good because it is the only way at the moment he can help, so he better do that well.
“We are looking at everything we could have done better, differently. But there are a lot of uncontrollable as well (with Saka’s injury).
“The action produces so much force because it is the moment he puts the cross in, so the force is in the standing legs and those actions happen in football.”