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United went into the match looking to end a largely disappointing season on a high, and it was Louis van Gaal's side who looked most like opening the scoring during a quiet start to the match.
Wayne Hennessey was called into action after quarter of an hour when Wayne Rooney, looking for his first FA Cup winners' medal, sent a deflected effort towards goal, but the Palace keeper collected it at the second attempt having spilled the original shot.
The main talking point of the first half arrived shortly afterwards, though, when referee Mark Clattenburg blew for a foul against Chris Smalling despite Connor Wickham having bounced straight back up and wanting an advantage as he broke through on goal.
Wickham proceeded to put the ball in the back of the net, but Clattenburg pulled play back for a free kick from which Yannick Bolasie flicked a header towards goal that needed to be tipped over by David de Gea.
Palace were beginning to settle into the game following a slow start, but they almost fell behind midway through the half when Mata steered a loose ball towards the bottom corner that Hennessey got down well to save.
The aerial battle between Fellaini and Jedinak was a running theme of the contest, and the United man got the better of the Palace skipper again in the 25th minute to reach a corner, but he couldn't connect with his header cleanly when well placed.
At the other end, Palace's sights of goals were being limited by United, with Bolasie drawing a stop from De Gea with a long-range strike moments before Wickham sent a 20-yard effort of his own wide.
It was United who were creating the better chances, though, and it took a last-ditch block from Joel Ward to deny Anthony Martial the opening goal shortly past the half-hour mark after good work from Marcus Rashford down the right flank.
Alan Pardew was left bemoaning another refereeing decision with 10 minutes left of the half when Wilfried Zaha robbed Daley Blind of possession and advanced into the area before going down under the challenge of a backtracking Rooney, only for Clattenburg to wave the penalty claims away.
The first half ended goalless, but United were inches away from breaking the deadlock eight minutes after the restart when Rashford flicked the ball through to Fellaini, who proceeded to power an effort against the post with Hennessey beaten.
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There was a hint of an opening for Palace shortly afterwards when Yohan Cabaye's free kick was flicked on towards the back post for Jedinak, but the Palace captain couldn't control his half-volley and blazed it well off target.
The Eagles' goal was living an increasingly charmed life, though, and United hit the post again shortly after the hour mark when Martial flicked Antonio Valencia's cross against the upright.
Any thoughts from Van Gaal that it might not be his side's day were increased with just 12 minutes remaining when Palace finally found the breakthrough, with Puncheon lashing a finish past De Gea from a tight angle just six minutes after coming on as a sub.
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It put Palace on the brink of a first ever major trophy in the club's 110-year history, but their jubilation soon turned to despair as United levelled things up again less than three minutes after falling behind.
Rooney created the chance by dancing his way into the right channel and hanging a cross up to the back post which Fellaini was able to chest down for Mata to volley past Ward on the line.
Both sides pushed for a winner to avoid extra time, and Palace had half a chance to get it when Pape Souare 's cross arrived to Zaha at the back post, but the former United winger failed to connect with his volley.
Zaha came close again in the fourth of five minutes added time when he muscled Smalling out of the way before rippling the side-netting, but there was no winner in the closing stages as the match went all the way to extra time.
Rooney sent one curling effort wide of the target in the additional 30 minutes, but it was Palace who came closest to regaining the lead when Bolasie's volley was turned round the post by a smart De Gea stop.
United suffered a setback in the final minute of the first half of extra time when Smalling became just the fourth player to be sent off in an FA Cup final, being shown a second yellow card for dragging down Bolasie.
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Palace looked to make the most of their numerical advantage early in the second half of extra time, and United needed De Gea to deny Dwight Gayle after the striker had collected Zaha's pass and poked one towards goal.
The Red Devils were not solely focusing on defence, though, and should have taken the lead with 10 minutes remaining when Michael Carrick arrived late into the box but could only steer his header narrowly wide.
They would not have to wait much longer for the winner, though, and it arrived in spectacular fashion as Lingard lashed a ferocious first-time volley into the top corner after the ball had broken to him just inside the penalty area.
Jedinak almost levelled things up immediately when he curled one just wide, before De Gea threw himself bravely at the feet of Zaha late on to keep the Eagles at bay.
Ultimately a first ever major trophy eluded Palace, however, as United ended a 12-year wait to win the FA Cup with their first piece of silverware under Van Gaal.