Dawber was a revelation alongside ALW Golden Boot winner Fiona Worts in Adelaide's attack, scoring 10 goals and notching three assists, as United finished third and made the 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."
"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."