James Poole's spectacular first-half strike booked Pools' place in the Northern Section quarter-finals and left the Blades, 22nd in League One, without a win since the opening weekend of the season.
Weir insisted he does not fear the sack and will work hard to turn things around. He held a lengthy post-match inquest with his players, who he admitted are short of confidence.
"I understand that results haven't been good enough and ultimately that's what I'll be judged on," said Weir. "It's unacceptable. We need to start playing better and start scoring goals and stop conceding goals.
"I'm a positive person and I look for the positives but I don't think there were any positives at all to take from the game.
"I don't think we created enough and that's obviously a big problem. There were many, many disappointments.
"With the run of results we've had, confidence is obviously an issue."