The 20-year-old has joined the Blades on a five-year contract for a club-record deal worth £23.5m.
The package for the England Under-21 forward includes a buy-back option for three years and the Reds will get 15% of any sell-on fee.
"He has been our number one target," said Wilder.
"At the top of the pitch we have one of the country's hottest properties from Liverpool and for him to sign permanently here is fantastic for everyone concerned.
"He's a natural goalscorer. He finishes from any aspect, whether it is in the air or whether it's spectacular goals, free-kicks or off his shin - he has a lovely habit of scoring goals, which is crucial in the Premier League."
"There were a few teams who came in but right now Sheffield United is the best option for me," said Brewster.
"Looking at the club, it is on the rise, finishing ninth last year and hopefully it can be better this season.
"Maybe I can add some goals - I'll work hard and try and help the team to a higher place this year.
"I am ready to come in and show everyone what I've got. It is a team that creates a lot of chances, the three at the back, overlapping, they get crosses into the box and that's my game - being in the box, finding space and one- and two-touch finishes.
"That's what I am going to try and do, and try and score goals for Sheffield United."