Keegan's appointment brings to an end days of speculation linking Mark Hughes with the Toon hot seat.
Rovers boss Hughes was immediately installed as one of the favourites for the position after Sam Allardyce left St James' Park last week.
Williams told the Lancashire Evening Post: "We're naturally relieved that it's sorted because our manager was continually linked with the post.
"I always felt that he would remain here. He values the continuity that he has and I know he feels we are close to a very good league position at the end of the season.
"We have a strong personal relationship and I never felt that he was likely to go to Newcastle.
"The jobs unfinished here and I'm sure that we will continue to make good progress.
"We're having a great season and the future looks good for us."