Oxlade-Chamberlain rejected a new deal worth £180,000-a-week at Arsenal and took a significant pay cut to join Liverpool in a £35million deal on deadline day.
The midfielder was set for a move to Chelsea, but turned down the opportunity as Antonio Conte had plans to play him as a wing-back to provide competition for Victor Moses.
And former Chelsea goalkeeper Schwarzer thinks Oxlade-Chamberlain made the right choice heading to Anfield.
“I think it’s a massive challenge for him,” Schwarzer told Sky Sports’ The Debate Show.
“At Arsenal he was a little bit stale.
“He became a little bit comfortable and frustrated he didn’t play in the position he wanted to.
“I’m actually really happy for him – it’s a great move for him and I think it’s the right move for him.
“I think it was probably as much as it pains me to say it, a better move than going to Chelsea.
“It suits him way he plays and Jurgen Klopp ideal for him.
“The high intensity of Liverpool and with the pace he has, he will play more of an attacking role.
“To me, he’s more exciting to watch in that more forward role than in a wing-back position.”