The result moves Chris Wilder's side to 11 points from 22 games -- 10 points away the safety zone and one behind 19th-placed West Brom, who have won only once in 10 games since Sam Allardyce took charge.
"When we're behind or winning we always go until the end because we know how quickly games can change. We never give up," said goalscorer Jayden Bogle.
Despite a lack of possession in the first half, West Brom went ahead in the 41st minute courtesy of Matt Phillips. The forward slotting home with a cool finish after Aaron Ramsdale parried a shot into path of the Scotland international.
Wilder's side struck back in the second half, defender Bogle sending the home side level in the 56th minute with a left-footed shot that struck past Sam Johnstone.
But local boy Billy Sharp secured victory for the Blades in the 73rd minute. Sharp managing to poke the ball into the back on the net from close-range to send the home side ahead.
"We needed a big second half performance. We huffed and puffed in the first half," Wilder said. "The second half was a lot better, more us. We were on the front foot and driving the opponents back.
"It was always not going to be a free-flowing game with the two teams in the situations they are in. We were just a bit slow in the first half, we just had to let ourselves go a little bit which we did in the second half. We were more ambitious."