The 34-year-old former Stoke defender made his Blades debut in the 2-2 draw at Doncaster on New Year's Day hours after signing a short-term deal until the end of the season.
"I got a phone call from Dave who had spoken to the gaffer, so I have got to be thankful for him for helping making this move happen," Higginbotham told the club's official website.
"Me and Dave go back along way and I didn't have to think twice when the deal was put on the table because Sheffield United is an ambitious club.
"Dave is a great player with a real touch of class. We have seen it at the top level for many years and he is a very clever footballer too.
"Hopefully he could be the difference in helping this club win promotion back to the Championship this season."
Higginbotham, who has played in the Premier League with Derby, Southampton and Sunderland, felt Danny Wilson's side showed true resolve to fightback from two goals down at Doncaster.
He added: "It wasn't looking like being a great start for me when we were 2-0 down, but the lads showed they have a real character and togetherness.
"It's a shame the game didn't go on for another 10 minutes or so because there was only going to be one winner."