Matt Phillips gave the Baggies a 41st minute lead before Jayden Bogle equalized in the 56th minute. Billy Sharp then broke the deadlock in the 73rd minute.
The win, Sheffield United's fifth in their last seven matches, puts the Blades on 11 points, one point behind West Bromwich Albion for 19th and ten points from safety.
Story of the game
West Brom started on the front foot with Jack Robinson, but on eight minutes, Bogle won a free kick after cutting inside on the left. Oliver Norwood took the attempt from 20 yards out only to fire high and wide.
After Robinson fired wide, David McGoldrick shook off two Baggies defenders to force a save out of goalkeeper Sam Johnstone. Ethan Ampadu then tried his luck, but Johnstone parried away.
More pressure from the hosts saw Norwood send in a pair of crosses with the West Brom netminder getting a fist to both and Kyle Bartley clearing the third try from John Fleck.
Against the run of play, the visitors took the lead four minutes from halftime. Bartley won the ball on the halfway line, Robert Snodgrass fed Robinson, who saw Aaron Ramsdale save, deny Mbaye Diagne on the rebound, but Phillips tapped home.
On 56 minutes, Sheffield United were level. Sharp's cross found Chris Basham, who advanced into the West Brom area, teed up Bogle and his left-footed shot beat Johnstone at his near post.
McBurnie and John Lundstram nearly gave the Blades the lead, but a couple of last-ditch blocks spared the visitors' blushes. Ramsdale then made the save of the game in the 63rd minuteB, denying Robinson from Phillips' low cross.
Sheffield United then produced the eventual match-winner as Fleck made space for Basham on the right. Egan rose above Andros Townsend and Sharp fired past Johnstone from ten yards out.
West Brom went hunting for a late equalizer as Matheus Pereira curled his shot wide from the edge of the area in stoppage time his set-piece was seen out as the Blades continue their revival.
Man of the match: Billy Sharp
He was a constant threat in the box and his finish was superb in what was his last action of the game. The key will be if the rest of the Sheffield United attacking players can provide him with the service needed for him to, as he has done so many times for the Blades, score crucial goals.
Sheffield United show fighting spirit
While staying up is still a monumental task, the Blades' recent run of form suggests they may still have a chance of mounting one of the greatest escapes in Premier League history.
Winners of three of their last five league games, Chris Wilder will be delighted that his side are showing their true potential and, more importantly, winning close games as each of those victories have come by a single goal.
West Brom still struggle to defend set pieces
Sam Allardyce will be furious that both of Sheffield United's goals came from crosses in the box that the Baggies failed to clear. For a manager that prides himself on having a strong defense, these failures will be both disappointing and alarming.
Having never been relegated from the top flight as manager and despite some good signings in the January transfer window, unless the West Brom defense improves drastically, that record will go by the wayside come the end of the season.