The Eagles appeared to have held on for a point despite playing the entire second half with 10 men following Dwight Gayle's brainless red card.
But West Ham's pressure paid off when Argentinian winger Lanzini rifled home in the 88th minute and Payet put the icing on the cake in injury time.
Carl Jenkinson had put the visitors ahead with his second goal in two matches, but Yohan Cabaye equalised moments later with a twice-taken penalty.
Payet had already threatened twice for West Ham before, in the 23rd minute, he slid the ball to former Palace winger Victor Moses, who fed overlapping full-back Jenkinson in the area.
The Arsenal loanee had sneaked in behind the daydreaming Wilfried Zaha and neatly tucked the ball past Wayne Hennessey with the outside of his right foot.
However, two minutes later Jenkinson was the central figure as Palace forced an equaliser.
The right-back and Gayle seemed to collide as they both stretched for the ball in the box, but referee Mark Clattenburg ruled in the Palace forward's favour and pointed to the spot.
Cabaye duly tucked away the penalty, only for Clattenburg to order a retake after spotting Gayle had encroached into the area.
Nevertheless, Cabaye held his nerve and lifted the ball high into the opposite side of the net despite Adrian getting a hand to it.
Gayle may have been breathing a sigh of relief at that stage but his next reckless act, a needless lunge on Payet, earned him a booking.
That proved costly for the striker when, in another rash challenge, he caught Cheikhou Kouyate late and was sent off a minute before half-time.
Not for the first time this season, Eagles boss Alan Pardew hauled off Zaha at half-time, while Andy Carroll was introduced shortly after as the Hammers attempted to make their numerical advantage count.
Diafra Sakho glanced Jenkinson's cross against a post but it appeared West Ham would head back to east London frustrated.
But when Carroll headed fellow substitute Mauro Zarate's cross back across goal Lanzini was on hand to smash the ball past Hennessey.
And in stoppage time Lanzini's through ball sent Payet clear and the Frenchman deftly chipped Hennessey to seal West Ham's fourth away win of the season.
TWEET OF THE MATCH
"Good to see Alan Pardew applaud the @whufc_official fans before kick off, a different view to the fans to Big Sam"
- Ian Abrahams @BroadcastMoose
https://twitter.com/BroadcastMoose/status/655382844274036736
PLAYER RATINGS
Crystal Palace
Wayne Hennessey: 6
Martin Kelly: 6
Scott Dann: 6
Brede Hangeland: 6
Pape Soure: 6
Yannick Bolasie: 6
Jason Puncheon: 6
James McArthur: 6
Yohan Cabaye: 6
Wilfried Zaha: 5
Dwight Gayle: 4
Subs
Bakary Sako: 6
Mile Jedinak: 5
Joe Ledley: 5
West Ham
Adrian: 7
Carl Jenkinson: 6
James Tomkins: 6
James Collins: 6
Aaron Cresswell: 6
Cheikhou Kouyate: 7
Mark Noble: 6
Manuel Lanzini: 8
Victor Moses: 6
Dimitri Payet: 7
Diafra Sakho: 6
Subs
Andy Carroll: 6
Mauro Zarate: 6
Nikica Jelavic: 6
STAR PLAYER
Manuel Lanzini - T he young Argentinian looks so composed on the ball and his all-round display was capped by a deserved late goal
Also found time to set up the third for Payet
MOMENT OF THE MATCH
Payet put the icing on the cake for West Ham with his fifth goal of the season, collecting Lanzini's pass and deftly lifting it over Hennessey.
VIEW FROM THE BENCH
Both managers use a 4-2-3-1, to good effect so far this season
The sides were cancelling each other out until Gayle's red card, allowing Slaven Bilic to bring on Andy Carroll and go with two up front
Palace almost held out but the pressure finally told.
MOAN OF THE MATCH
Dwight Gayle - A striker who has worked so hard to get into Palace's team promptly took himself out of it with two reckless tackles before half-time
Almost cost his side further before that, having encroached when Cabaye took his first penalty.
WHO'S UP NEXT?
Leicester v Crystal Palace (Barclays Premier League, Saturday October 24)
West Ham v Chelsea (Barclays Premier League, Saturday October 24)
Source : PA
Source: PA