Dawber was a revelation alongside golden boot winner Fiona Worts in Adelaide's attack, scoring 10 goals and notching three assists, as United finished third in the ALW and made finals for the first time in their history.
The 22-year-old, who is yet to be capped, has joined Chicago on a two-year deal, with an option for a third.
"It's a new adventure for me and one that I'm really excited about," Dawber said.
"I want to thank Adelaide United and all the coaching staff I've worked with for believing in me and giving me the opportunity to play for this amazing club.
"My goal has always been to play for the Matildas and I believe playing overseas in a tough league will give me a better chance of a call-up to the national team.
"I'm definitely not ruling out coming back to Adelaide after the NWSL season.
"Approaching a World Cup year, it's vital to be playing as many games as possible and give myself the best path for success."
Chicago was previously the home of Matildas captain Sam Kerr, who shot to global prominence with standout goal-scoring performances for the Red Stars in 2018 and 2019 before joining Chelsea.
"Chelsie is an impressive young player. She has proven she can score goals at a high level, play multiple positions and still be effective," Red Stars coach Chris Petrucelli said.
"She adds versatility and athleticism to an already dangerous attacking group."
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