The 24-year-old joined Liverpool in a £35m ($48.25m) deal on transfer deadline day last summer after turning down a £180,000-per-week deal to remain at the Emirates Stadium.
Oxlade-Chamberlain has gradually improved his displays for the club with arguably his finest moment coming in Liverpool's 4-3 win over Manchester City where he scored the opener en route to ending Pep Guardiola's unbeaten league record this season.
The pressure to perform will only increase for the England international following the big-money sale of Philippe Coutinho and Wright believes he will take his chance with both hands, having had to fight for his place unlike at Arsenal.
"He has to perform and get it done, and that is what Klopp has said to him - 'this is your chance'. I feel there has be times at Arsenal where you can see players who to me look like they're not motivated enough," Wright said, as per Liverpool Echo.
"They know they're going to play and when they do play, they are not sure how important it is to them if they lose, or if they don't play well.
"You get the impression that he realises now that it was a magnificent opportunity he has taken for himself."
Oxlade-Chamberlain was regarded as a transfer flop with his first few performances for the Reds, especially having debuted in an embarrassing 5-0 loss to City back in September.
But Wright states that all his critics will go away if he continues to perform the way he has in recent games.
"A lot of people teased him," he added. "There were lots of GIFs and memes coming out over his first few games.
"At some stage some of those people will have to shut up and hopefully that will happen for him.He made a very difficult decision for himself and I'm pleased at this particular moment in time it is working out for him."