Barnsley have survived relegation from the Championship for another year following their creditable 2-1 one over Plymouth at Home Park.
Norwich City's defeat at the hands of Charlton means the Tykes would have beaten the drop on the final day of the season, regardless of their comeback against Plymouth who were already safe.
Gary Sawyer's header opened the scoring on 11 minutes to set up a frantic relegation scrap. Honours were even at the break though as on loan midfielder Adam Hammill levelled for Barnsley on 34 minutes to send the Tykes on their way to safety.
Jamal Campbell-Ryce then put the away side into the lead just one minute into the second half to send the away fans delirious, with survival well within sight.
Simon Davey's side held out their lead until the final whistle, leapfrogging Plymouth into 20th position in the process.
Pace and purpose
The Clarets attacked with pace and purpose in the opening quarter with the outstanding Chris Eagles at the heart of all their most promising moves.
The former Manchester United winger illustrated his quality in the eighth minute by flicking up the ball from Elliott's pass and unleashing a fierce right-foot shot from around 10 yards.
City goalkeeper Adriano Basso was well beaten and grateful to see the ball rebound off the crossbar to safety but the visitors, with nothing but pride left to play for, gradually gained a foothold on proceedings.
Striker Nicky Maynard had the ball in the net in the 11th minute but an offside flag had already been raised.
Yet Burnley remained largely dominant and led from the spot in the 42nd minute after Jamie McAllister clumsily fouled Elliott.
Alexander produced a perfect penalty as he fired the ball into the top left corner of Basso's goal.
A second goal arrived two minutes later when City failed to clear their lines and the ball fell to the unmarked Elliott, who rifled home a fine left-foot shot from 20 yards.
Burnley were rarely troubled after the break as City's heart for the fight gradually evaporated.
The Clarets' enterprise was rewarded with a third goal in the 76th minute when a cross from Eagles was handled by Robins right-back Bradley Orr.
Alexander stepped up again to fire high into the net and Turf Moor erupted once more with four minutes remaining when Eagles' shot was saved by Basso and Gudjonsson tucked away the rebound with a close-range strike.