Matty Robson's 54th-minute header was enough to secure Carlisle their first league win of the season. It came when the Blades' defence failed to deal with James Berrett's corner and although the ball only just crossed the goalline before it was scrambled away, the referee's assistant flagged for the goal.
The Blades weaved pretty patterns, but were punchless and the result left Weir saying: "I'm disappointed and frustrated and questioning a lot of things."
After a lengthy post-match inquest with his players, the former Everton coach said: "We are trying to get to the root of the problems. We will try to be honest with each other and improve, but it's hurting us all.
"We aren't getting the end results. We are not creating enough chances or scoring enough goals and conceding too many.
"There are a lot of questions to answer and it is a big challenge for us. When you get lows like this you can back down or stand up and face it, and I'm the sort of person to stand up and face it.
"It's a real low for us, we can't get away from it. I am questioning myself, but the answer is not walking away but to look for answers."
Substitute Callum McFadzean sparked a late response fromthe Blades and his low cross from the left skidded across the six-yard box. The winger then laid on a chance for skipper Michael Doyle, who lifted his shot high over the crossbar from 12 yards.
But Carlisle held on well for their first home clean sheet of the season and Troy Archibald-Henville nearly made it safe when he skewed a close-range header wide in the dying minutes.
While the Cumbrians had their first three-pointer of the season to celebrate, the result left Blades without an away win and slipping below Carlisle in the table.