It looked so good for the Reds when Jay Donnelly shot them into a first-half lead.
But Andy Waterworth levelled after the break before Burns produced his party piece in extra-time - inflicting the first defeat on new Reds boss Gerard Lyttle.
The Blues were almost gifted a lead in the second minute. George McMullan slipped as he picked up a clearance from goalkeeper Peter Burke, leaving Stephen Lowry with a gaping goal.
But Burke got back to paw the ball to safety.
That sparked the Reds into life. Defender Levi Ives whipped in a cross from the left which was met by Jude Winchester, who headed yards wide.
But the Blues threatened again with Reece Glendinning's impressive solo dash cutting open the Reds at the back, but instead of pulling the trigger, he tried to pull back to Guy Bates, who was yards off the pace.
Two minutes later the gangling Bates attempted an audacious 25 yard lob after out-smarting Johnny Flynn. His effort landed on the top of the net.
Ivan Sproule thought he had given the Blues the lead seven minutes from the break.
Lowry's sent the former international clear and, after rounding Burke, he finished in style. Referee Mervyn Smyth, however, ruled out the effort, saying Sproule had strayed offside.
It was the away fans who were celebrating four minutes before the interval. Tomas Cosgrove roared through a challenge from Niall Quinn before leaving Sproule for dead and his cross was finished by Donnelly.
But the Blues wasted a wonderful chance to draw level 60 seconds later.
Jamie Mulgrew was hauled down by a late tackle from McMullan, a clear penalty kick.
Waterworth stepped up only to see his low drive saved.
Linfield required only seven minutes of the second-half to draw level. Quinn picked up the scraps after the Reds conceded possession and, when he drilled in a cross from the left, Waterworth's header crashed past Burke's outstretched arm.
The visitors were almost in front again on 59 minutes. James Knowles' corner-kick was met by Cosgrove, but his header hit the crossbar.
Lowry was Linfield's saviour two minutes from time when a lightning break presented David McDaid with a chance. He looked a certain scorer until Lowry dived in.
Linfield eased in front three minutes into extra-time. Kirk Millar's corner was met by Waterworth, whose shot was edged over the line by Burns.
Burke then pulled off a sensational save to repel a rasping drive from Millar as the Blues went for the kill.
Healy's men sealed it in the last minute of extra-time. Bates tumbled under a challenge from Ryan Catney and the referee once more pointed to the spot. This time, Burns made no mistake.
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