The visitors took a deserved lead after 13 minutes through Callum Wilson.
The Blades are now two points behind Chelsea, albeit with a game in hand.
Eddie Howe's Bournemouth are one point above the relegation zone in 16th, but teams below them have a game in hand.
Speaking after the game, Wilder said: "We were hanging on at times. We grew into the game and I felt a goal was coming. I was delighted to go into the break on level terms.
"We try to build momentum and create pressure. They controlled the game early on but our reaction was great. We really went for it in the second half and the belief the boys have is amazing.
"I'm sure when we were coming off our fans were singing that we're safe. We're not there yet but there's no let-up from us. There are no comfortable footballers in our changing room - they all have something to prove.
"I'm a bit controversial in that I'd want to carry on - I'd like it if we had a game next week. But we'll come back refreshed for the next part of the season."