Palace were on course for a shock victory after local hero and substitute Jason Puncheon fired them ahead with just 12 minutes remaining but Wayne Rooney set up an immediate Juan Mata equaliser.
United lost defender Chris Smalling in extra time after he hauled back Yannick Bolasie to incur a second booking but United snatched victory with a fine volley from substitute Lingard after 110 minutes.
Defeat was harsh on valiant Palace, who were frustrated by a series of decisions by referee Mark Clattenburg in the first half and fought to the end.
A touchline dance from Alan Pardew after Puncheon struck suggested they sensed history and revenge for defeat in the 1990 final, but their jubilation was shortlived.
United could argue, after hitting the post twice and bossing large spells, they deserved victory and a first trophy since the retirement of Sir Alex Ferguson.
But no matter how it came, it may do little to end continuing speculation over the future of their manager Louis van Gaal. The Dutchman again woke on the morning of the match to fresh reports linking former Chelsea manager Jose Mourinho with his job and such stories may only intensify.
They match kicked off late following a jazzed-up opening ceremony that did not run to time - or plan, with national anthem singer Karen Harding missing her cue.
But the noisy Palace fans were eager to make up for that, with their end of stadium being awash with flags, balloons and colour, making for a fervent atmosphere.
Yet United settled quicker, winning a succession of corners in the early stages and Marouane Fellaini, returning after suspension, twice headed wide. Mata also forced a fine save from Wayne Hennessey.
Palace found it harder to gain a foothold but Clattenburg infuriated them by not allowing Connor Wickham advantage after he broke clear of Smalling following a tussle, instead awarding them a free-kick. They were similarly frustrated when Joel Ward was called back after shaking off the attention of Marcos Rojo but, tellingly, Smalling was booked for his misdemeanour.
Palace also wanted a penalty after Wilfried Zaha robbed Daley Blind and then went down in the area under a Rooney challenge but replays suggested that decision could have gone either way.
United upped the tempo early in the second half and twice hit the woodwork with a fierce shot from Fellaini and glancing header from Anthony Martial.
The game went quiet for a spell but exploded into life as Palace grabbed the lead against the run of play through Puncheon.
The boyhood Palace fan had been on the field just six minutes when the ball was swung back into the area by Ward. Puncheon, just onside, took a touch and then hammered a fierce shot past David de Gea to spark jubilant celebrations among the Palace fans behind the goal.
Palace boss Pardew joined in the fun but their delight did not last long as, just four minutes later, Rooney wriggled his way to the byline and lofted a cross into the danger area. Fellaini chested down and Mata ran in to equalise.
Palace may have feared a late onslaught but that did not arrive Wilfried Zaha went the closest to a normal time winner when he hit the side-netting after robbing Smalling.
In extra time Lingard forced Pape Souare to head narrowly over his own bar before Bolasie tested De Gea with a firm volley from outside the area.
United’s task got harder just before the break in injury time as Smalling was turned and beaten by Bolasie. His attempts to stop him were akin to a rugby tackle and Clattenburg had no option but to show a second yellow card.
De Gea came to United’s rescue as he saved with his feet from Dwight Gayle before Michael Carrick headed a good chance wide at the other end.
Yet United’s moment of glory was still to come as Lingard lashed in a volley from the edge of the area after a cross was half-cleared by Damien Delaney.