Norway international Sander Berge gave the home side the lead after the break when he was teed up by Iliman Ndiaye following an awfully under-hit backpass from Taylor Gardner-Hickman.
The Blades, who went into the game needing just three points from four games to secure promotion, had been sluggish up until then but overwhelmed the away side from thereon and Anel Ahmedhodzic scored with a controlled close range volley to seal victory with just under 15 minutes to play.
Paul Heckingbottom's side will join title winners Burnley in the top flight next season.
The play-off chasing visitors had enjoyed the better of things up until the opening goal but Karlan Grant headed a great chance wide in the opening minutes and a penalty shout was turned down when Brandon Thomas-Asante was felled on the edge of the area.
Berge's goal totally knocked the stuffing out of Carlos Corberan's men though and the Blades' margin of victory could have been more comprehensive at the end.
West Brom, who were playing their game in hand on the sides around them, remain two points outside the top six with matches against play-off rivals Norwich and Swansea to finish the campaign.
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