The Tykes suffered a 4-1 defeat at Derby County which, combined with Bolton's dramatic 3-2 win against Nottingham Forest, saw their two-year tenure in the Championship come to an end.
Burton, meanwhile, appeared safe heading into the latter stages of their game at Preston North End, as they were level against the Lilywhites.
Bolton's remarkable comeback, however, combined with Louis Moult's late strike for the hosts, saw Nigel Clough's side fall to a 2-1 defeat and drop to the third tier alongside Barnsley and Sunderland.
Higher up the table, Derby claimed the final playoff spot, finishing ahead of Preston, whose win over Burton was not enough to see them over the line.
The Rams join Middlesbrough, Aston Villa and Fulham - who fell to a 3-1 defeat at Birmingham City to gift second place to Cardiff City - in the playoffs.
Championship results in full: Birmingham City 3-1 Fulham, Bolton Wanderers 3-2 Nottingham Forest, Brentford 1-1 Hull City, Bristol City 2-3 Sheffield United, Cardiff City 0-0 Reading, Derby County 4-1 Barnsley, Ipswich Town 2-2 Middlesbrough, Leeds United 2-0 Queens Park Rangers, Millwall 1-0 Aston Villa, Preston North End 2-1 Burton Albion, Sheffield Wednesday 5-1 Norwich City, Sunderland 3-0 Wolverhampton Wanderers