United fell behind to substitute Jason Puncheon's 78th-minute strike, but Juan Mata levelled within 180 seconds to take the game to extra time.
Chris Smalling – earlier embroiled in controversy after hauling back a goal-bound Connor Wickham – saw red just before the end of the first additional 15 minutes following a foul on Yannick Bolasie, but substitute Lingard proved the Reds hero as they etched their name on the famous trophy for a record-equalling 12th time and first since 2004, casting aside pre-match speculation that Van Gaal is to be replaced by Jose Mourinho.
The Dutchman's failure to steer United into the Champions League is a key catalyst for his likely exit, but the 64-year-old can at least boast a major honour from his two-year tenure after Saturday's deserved victory.
However, it was heartbreak for Palace and their boss Alan Pardew, who played when the club last reached the Cup final – United prevailing back in 1990 after a replay – and managed West Ham in their 2006 penalty shootout loss to Liverpool.
United controlled the first half and hit the woodwork twice in the second period through Marouane Fellaini and Anthony Martial, before Puncheon stunned them with a fine near-post finish.
But the London club's joy was cut short, Mata's deflected volley squeezing through Joel Ward's legs on the line in the 81st minute before Lingard – on for the Spanish goalscorer in stoppage time – was cast in the role of match-winner in a result that sent the Hammers into the Europa League.
Van Gaal welcomed back Fellaini from suspension, with Mata handed a start, while Palace returned to a more familiar starting XI after making numerous changes for the 4-1 loss to Southampton.
United, who finished the Premier League season later than scheduled on Tuesday after the bomb scare prior to their final game against AFC Bournemouth, made a bright start following a kick-off delayed by the pre-match entertainment.
Wayne Hennessey twice claimed excellent catches from searching crosses to calm any nerves, while Marcus Rashford – who hobbled off injured at the end of a week in which he was named in England's provisional Euro 2016 squad – forced a timely intervention from Pape Souare after a strong run and shot.
Palace sprung forward on a rare 18th-minute counter-attack and saw Wickham pulled back by Smalling, with Pardew livid at referee Mark Clattenburg's decision to stop play as his forward had managed to shrug off the defender and roll the ball into an empty net after the whistle had gone.
David De Gea tipped the subsequent free-kick over the crossbar and Mata almost rubbed salt in the wounds with a skidding effort that Hennessey clawed away midway through the half.
A deft turn from Rashford had Palace chasing back in desperation as the teenager's cross was steered goalwards by Martial and Ward made the block.
Ex-United winger Wilfried Zaha saw his run into the box halted by a recovering Wayne Rooney, the United captain proving his value at both ends of the field in a deeper role that looks set to become his forte.
Fellaini – who scored in the semi-final win over Everton – served notice of his intent early in the second half, rifling a shot against the frame of the goal and past a motionless Hennessey, who saw Martial's header strike the left-hand post just past the hour mark.
Rashford was forced off 20 minutes from time and replaced by Ashley Young, while Palace swapped Yohan Cabaye for Puncheon, whose one-touch control and fierce finish broke the deadlock to prompt a touchline jig from Pardew.
His apparently rehearsed moves were quickly put on the back-burner, though, as Mata wiped out that lead, though Zaha – pictured in tears prior to the game – could have snatched a winner when he fired into the side netting at the death.
Bolasie drilled wide in the first half of extra time, while Smalling got his marching orders just before the break, but United shrugged off that setback as Lingard lashed home brilliantly from 18 yards in the 110th minute to salvage an otherwise uninspiring season for the Old Trafford club.