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Goalkeeper Emiliano Martinez was the hero in the shootout as La Albiceleste edged past the Oranje 4-3, after an absorbing 2-2 draw at Lusail Stadium.
Martinez produced saves from Virgil van Dijk and Steven Berghuis before Lautaro Martinez slotted home the winning spot-kick as Argentina secured a semi-final berth against Croatia, which earlier in the day knocked out Brazil on penalties.
Messi looked to have won it for Argentina in normal time by creating Nahuel Molina's opening goal in the first half before adding another himself from the penalty spot, only for two late Wout Weghorst efforts to send the game beyond 90 minutes.
A low-key first half in which both teams were content to keep possession without being good enough to create anything with it was finally ignited by the brilliance of Messi 10 minutes before the break.
The Paris Saint-Germain star picked the ball up 40 yards from goal, took a couple of players out with a body swerve and then slid it through to Molina who finished well from 10 yards.
5 - Lionel Messi has assisted five goals in the World Cup knockout stages - since Opta have World Cup assists (from 1966), this is the most on record in the knockout rounds of the finals, surpassing Pelรฉ's four. Goat. pic.twitter.com/ScHbh5oj1b
โ OptaJoe (@OptaJoe) December 9, 2022
The second goal for Argentina came on 73 minutes after a senseless trip by Denzel Dumfries on Marcos Acuna just inside the penalty area.
Messi, who missed from the spot against Poland earlier in the tournament, made no mistake this time, wrong-footing Andries Noppert to extend his side's advantage.
Netherlands boss Louis van Gaal introduced Weghorst with 12 minutes to go and it proved to be a masterstroke.
The tall striker Weghorst halved the deficit five minutes after his introduction, heading home fellow substitute Berghuis' in-swinging cross, and landed another telling blow with virtually the last kick of normal time.
With everyone in the stadium expecting Teun Koopmeiners to go for goal with a free-kick 20-yards out, he instead slipped a clever ball through to Weghorst and he did the rest, sliding a left-foot shot past Martinez.
๐ณ๐ฑ v ๐ฆ๐ท โฝ๏ธ 90+11' Incredible. @OnsOranje has equalised in stunning fashion with super-sub Wout Weghorst slotting home after an audacious free-kick caught @Argentina by surprise. #NED 2-2 #ARG | Follow it LIVE HERE ๐ https://t.co/GsfF6noOyX#FIFAWorldCup #NEDARG #Qatar2022 pic.twitter.com/v83DvvYGlI
โ beIN SPORTS (@beINSPORTS_AUS) December 9, 2022
A match that had seen regular flashpoints in the second half boiled over again after the final whistle with players from both sides embroiled in pushing and shoving.
Argentina pressed for a winner in extra-time but could not find a breakthrough, with Enzo Fernandez coming closest in the final minute with a 20-yard drive then crashed against the post.
Martinez then took centre stage, saving the Netherlands' first two efforts to put Argentina in control.
Enzo Fernandez put his effort wide to increase the tension but Lautaro Martinez made no mistake, firing home to spark wild celebrations.