The American Idol judge was asked whether he thought Simon Cowell had been happy with The X Factor's first season during an appearance on Jimmy Kimmel Live on Tuesday night.
Jackson replied: "I think he was. At the end of the day they had somebody who was great that won, she's a talented girl. I watched the finale and she's really talented. She's a great young singer from Florida.
"You look at these shows as we do on Idol... and when you find these artists and they win the show they have to go on and sell records. Everything you have been preaching and saying, 'This is the reason you should win', things have to happen.
"I think I expect big things for that girl."
Jackson also gave a shout-out to American Idol's season ten winner Scotty McCreery, whose debut album Clear As Day has just been certified platinum with more than one million copies sold in the three months following its release.
As for the supposed rivalry between American Idol and The X Factor, he said: "Of course you always want to remain the number one show but it's friendly fire. [Simon] will call us with stats, Ryan [Seacrest] and I will call him with things.
"It's what we are. We were the three musketeers at that time, travelled together, hung together, ate together, the whole thing."
American Idol returns for its 11th season with a two-night premiere featuring audition highlights from Savannah and Pittsburgh on Wednesday, January 18 and Thursday, January 19 at 8/7c on Fox.
Jackson added: "We've finished auditions, finished Hollywood Week... America is just full of a lot of talent and we found some amazing kids this time."
> American Idol judges: 'There are a lot of frontrunners this season'
> Read Digital Spy's full American Idol headlines
Watch Randy Jackson talking to Jimmy Kimmel below: