Marcelo Bielsa's side went ahead in the tenth minute as Harrison finished off a fine move after Mateusz Klich's attempt had been saved.
The Whites then lost both Adam Forshaw and Junior Firpo to injury around the quarter mark and the Irons drew level 11 minutes before the break when Jarrod Bowen headed home a corner.
United went back ahead just three minutes later as Harrison doubled his tally by netting from close range from a Raphinha corner.
The Irons levelled for a second time in the 52nd minute as Michail Antonio's flick released Pablo Fornals who cut inside and beat Illan Meslier with a low finish.
Yet Leeds went back ahead on the hour mark as Harrison completed a superb hat-trick when slotting home after Raphinha's through ball.
The Whites were then denied a fourth as Klich's strike was disallowed by VAR for offside after hitting Rodrigo practically on the line but Harrison's treble was enough for a memorable success.
Graham Smyth, the YEP's chief football writer, hands out the scores from a memorable afternoon at the London Stadium.
7. Adam Forshaw
7 - Was playing well when he went off injured. A troubling one for Leeds given his importance in recent weeks. Photo by Mike Hewitt/Getty Images.
8. Mateusz Klich
7 - He and Forshaw were ticking along nicely in the opening stage. Couldn't keep hold of the midfield but he never stopped working to disrupt and attack. VAR robbed him, too. Photo by Mike Hewitt/Getty Images.
9. Jack Harison
9 - For a player who has struggled for end product this season he was deadly. Third finish was delightful. Worked hard and created danger too. Photo by Alex Pantling/Getty Images.
10. Raphihna
9 - Dangerous when he got the ball. West Ham really struggled to contain him when he was running with it. His pass for Harrison was pure class. Photo by DANIEL LEAL/AFP via Getty Images.
11. Dan James
7 - Decision making and final effort eluded him at times but he pressed throughout and gave West Ham difficulties. Photo by DANIEL LEAL/AFP via Getty Images.
12. Leo Hjelde (sub)
6 - Solid if unspectacular. Did his job, got stuck in. A couple of nice moments on the ball. Photo by Alex Pantling/Getty Images.