Defeat leaves Boro four points adrift of their neighbours, Sunderland, in the race for the second automatic First Division promotion place, but the gap must have looked even wider when Paul Gascoigne, the man Robson bought from Rangers to lead the end-of-season charge, limped off in the 77th minute.
A badly bruised and gashed left foot is likely to keep him out of Middlesbrough's next two games, though that may be something of a relief for Gazza, who has now finished on the losing side in all three matches he has played since his move.
Bramall Lane saw the best and worst of him. The best was a 20-yard surge down the middle and a cross that none of his team-mates could meet, followed by a neat turn to set up Paul Merson, but, in a foretaste of things to come, the chance was scooped over the bar.
The worst was a peevish spat with Ian Hamilton which could have earned him his third successive booking.
Saunders' goal lifted the Blades to fifth in the First Division and put Wembley, via the play-offs, back on their agenda.
Their manager, Steve Thompson, was less than happy that the needs of television forced his side to face another crucial challenge from the North-east just 48 hours after losing to Newcastle, but his fears were misplaced. The Blades looked much the sharper, and if their finishing had matched their neat approach work they would have wrapped it up by half-time.
Saunders wasted two clear chances before eventually finding the net. Steve Vickers conceded a free-kick by the corner flag, Wayne Quinn floated it in, the Middlesbrough goalkeeper Mark Schwarzer could only parry Lee Sandford's header, and Saunders was lurking to bundle the ball in.
Yet Middlesbrough could still have come back. Their Colombian striker Hamilton Ricard shot wide after Alun Armstrong had put him clear and then won a penalty when Hamilton brought him down, only to see Merson scoop his spot-kick yards over the bar.
"That was my worst moment in football for years," Merson said. It was not much better for Robson or Gascoigne, who will not play again until after Glenn Hoddle names his England squad for the friendly with Portugal.
Sheffield United (4-4-2): Kelly; Borbokis, Holdsworth, Sandford, Nilsen; Stuart, Hamilton, Ford, Quinn; Taylor (Marcelo, 85), Saunders (Devlin, 85). Substitute not used: Dellas.
Middlesbrough (4-3-1-2): Schwarzer; Fleming, Vickers, Festa, Harrison; Maddison, Mustoe, Gascoigne (Ormerod, 77); Merson; Armstrong, Ricard. Substitutes not used: Kinder, Branca.
Referee: E Lomas (Manchester).Reuse content