The midfielder joined the Gunners from Hammarby for an initial fee of around £140,000, although it is believed that figure could rise to £250,000 with bonuses.
Arsenal were aware of that and made a decisive offer to Hammarby on Thursday before the summer transfer window closed in the WSL.
The 21-year-old was one of the stars of the World Cup this summer, helping co-hosts Australia to reach the semi-finals on home soil, and her arrival rounds off an impressive window for the Gunners.
Arsenal also had a bid for England goalkeeper Mary Earps rejected by Manchester United, but they will continue to monitor the Lionesses star as her contract expires at the end of this season.
Signing Cooney-Cross is part of Arsenal’s push to bring in young talent and she joins an exciting crop of midfielders that includes Frida Maanum, Kathrine Kuhl and Victoria Pelova, with a plan for them to learn from experienced pair Kim Little and Lia Walti.
“Kyra is one of the best young players in world football and we are delighted to have brought her to Arsenal,” said Arsenal head coach Jonas Eidevall.
“Her passing ability and explosiveness allow her to affect games in a way you rarely see from such a young player, so we firmly believe she will be an excellent addition to our midfield, now and in the future.”
Cooney-Cross added: “I’m so excited to join this club. Arsenal just felt right as soon as I heard about the interest.
“I’m really looking forward to experiencing the atmosphere at Emirates Stadium in front of a big crowd for our first game there this season. I can’t wait to get started and help us achieve something big together.”