The former defender believes Arsenal's double triumph the year before spurred Alex Ferguson on to spend big and ensure United reigned supreme domestically with an FA Cup win and Premier League glory.
That investment in the squad served them well in Europe as the Red Devils went on to claim a famous 2-1 victory over Bayern Munich in the 1999 Champions League final to wrap up a treble.
Answering a Q&A, Keown told The Daily Mail: "When Man United won the treble in 1999 that almost had as much to do with us winning the double in 1998.
"Man United went out and spent big money on big players because they wanted to dominate domestically and that spurred them on.
"You have to be dominant in your own back garden before you challenge elsewhere, and if your neighbours are pretty decent you push each other on.
"In my era it was all about taking away the dominance of the other club.
"By the time of the invincibles, we were doing things they had never done before.
"It pushes you on because you know you can’t let up.
"Every week you have to at least match the other team if not better their result and you drag the best out of each."
