The opener came in the 22nd minute when Chelsea loanee Victor Moses found a perfect pass for Jenkinson to run onto with the Arsenal right-back controlling well before toe-poking beyond Wayne Hennessey with a composed finish for his second in two games.
Palace responded almost immediately when Jenkinson caught Dwight Gayle in the area which allowed Yohan Cabaye to step up the spot and fire beyond Adrian despite having to re-take his initial penalty when Gayle had encroached the line.
The home side's job became a little more difficult when Gayle was sent off on the stroke of half-time for a second bookable offence.
With a one-man advantage, West Ham searched for another go-ahead goal and came up with the answer in the 88th minute when Lanzini rifled home from close range before Payet wrapped it up in injury-time when he received a lovely ball from Lanzini before deftly chipping over Hennessey for his fifth of the season.
- West Ham are one of the in-form teams in the league. Their away form can push them towards a top-five finish if they can avoid unnecessary home defeats.
- Palace are mixing their form of late. If they find consistency, can they push the top five or will they drop off as the season progresses?
Hey Hammers , A big thank you for your kind messages and the support I really appreciate.
— Dimitri Payet (@dimpayet17) October 18, 2015
JENKO: @carljenkinson says the Hammers have 'immense confidence' at present. More at http://t.co/7KQbdaQrFe#WHUFCpic.twitter.com/omTZApo4Q2
— West Ham United FC (@whufc_official) October 18, 2015
Just wanna apologise to the team and the fans
— dwight gayle (@dwightgayle) October 17, 2015
Palace's Pardew on Gayle red:
"The first challenge from Dwight was bad and was a yellow card.
"But the second, he's missed the ball by about a centimetre. Perhaps he shouldn't have made the challenge, and he will have to learn from that, but I still think the referee was hasty."
West Ham boss Bilic on on getting over 10-man Palace:
"We had to be patient and stay in position. It's not easy to score against a team basically defending with nine men.
"I was more pleased with performance in the first half. We came here with respect for Palace, they are doing well."
*Thanks to Opta
15 - Dimitri Payet has scored or assisted 15 goals in his last 17 league appearances. Chipper.
— OptaJoe (@OptaJoe) October 17, 2015