The 26-year-old striker was immediately rushed to hospital for treatment.
Dutch national coach Bert van Marwijk conceded the situation did not look good, and on Sunday evening the Dutch FA (KNVB) confirmed van Persie had suffered ligament damage.
However, the outlook looked more promising when it became clear that surgery would not be required.
Van Persie later told Dutch television programme Studio Voetbal: "I have had a scan.
"I am probably out for between four and six weeks."
He did, however, absolve Juventus defender Chiellini of blame, adding: "There was no intent."
Van Persie has scored seven goals in 11 Premier League appearances so far this season.
His injury comes at a bad time for Gunners boss Arsene Wenger, whose attacking stocks have also been reduced by Nicklas Bendtner's groin injury, which he picked up in the recent 3-0 north London derby win against Tottenham.
Bendtner is expected to be out for at least another two weeks.
A four-week absence would rule van Persie out of the forthcoming league fixtures against Sunderland, Chelsea, Stoke and Liverpool, as well as the Gunners' final two Champions League Group H fixtures and the Carling Cup quarter-final at Manchester City.
However, he could feasibly be available by the time Arsenal travel to Burnley on Wednesday December 16, while Hull, Aston Villa and Portsmouth provide the opposition in the subsequent fixtures over the Christmas period before the FA Cup third round on Saturday January 2.
Arsenal 1/2, Draw 3/1, Sunderland 6/1