Arsenal suffered a 1-0 defeat to Liverpool in their first Women's Super League game, despite speculation they could be title contenders this season.
But Eidevall - who earned the club their first silverware in four years - remains convinced of their potential.
He said: "We've made great progress on and off the pitch over the last couple of years and we are all working hard to continue that growth every single day.
"Our ambition is to be competing for the highest honours in women's football consistently and we all believe we are capable of achieving that.
Eidevall made the move from Swedish side FC Rosengard in June 2021 and led Arsenal to a second-place finish in the WSL in his first season in charge.
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They then managed a third place finish last season in the WSL, also making it to the quarter-finals of the Champions League and winning the Continental Cup title.
A new record attendance for a WSL match was set at Arsenal's last match at the Emirates as 54,115 turned up to watch their defeat to Liverpool.
Eidevall sent a message of gratitude to fans for their continued support across all competitions.
He said: "I'd also like to say a special thank you to our supporters. From the moment I arrived, I've felt your warm support and it means the world.
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"Home or away, our supporters never fall short - to represent this club, this community, is an honour. Together, we can achieve so much."
But there will be no repeat of their Champions League success this season after they were knocked out in a shock defeat to Paris FC in the qualifying rounds.
Arsenal's focus now turns to the WSL which has been dominated by Emma Hayes' Chelsea for the last four seasons.
Arsenal's Sporting Director Edu Gaspar said: "This is an exciting time for our women's first team as we approach a new season and this announcement is an important sign of our renewed commitment to the journey ahead.
"As a club, our aim is to be challenging at the top of the game and Jonas will play a key role in us achieving that."
Arsenal will have the chance to revive their season as they travel to Leigh Sports Village to face last season's runners-up Manchester United on Friday October 6 at 7:30pm.
Manchester United came from behind to beat Aston Villa 2-1 in their opening game and will hope to continue their winning ways if they are to challenge Chelsea for the top spot this season.