Premier League: West Ham 0 Crystal Palace 1
Palace completed a double over their fellow Londoners, having triumphed by the same scoreline at Selhurst Park earlier in the campaign, with a fifth consecutive top-flight win – a run on a par with their previous best mark, set in December 1992.
Jedinak, an ever-present in the league this season, netted his maiden Premier League goal with aplomb after Pablo Armero had fouled Cameron Jerome.
Until that point, West Ham had crafted the clearer openings but centre-forward Andy Carroll passed up chances to press his claims for a place in England's FIFA World Cup squad.
Palace are now mathematically safe from relegation with three games of the season remaining, while West Ham should have already done enough as they lie seven points clear of the bottom three.
Before kick-off, Upton Park paid poignant tribute to young West Ham forward Dylan Tombides – the 20-year-old who died on Friday following a three-year battle with testicular cancer.
Tombides wore squad number 38 and fans later stood to applaud throughout the 38th minute of the match.
West Ham captain Kevin Nolan, back from the hamstring strain that kept him out of the midweek defeat at Arsenal, hooked an eighth-minute cross to the back post where Carroll headed over under pressure from Adrian Mariappa.
Carroll was soon involved at the other end, clearing off the line to deny Kagisho Dikgacoi a first Premier League goal from Yannick Bolasie's corner.
Mariappa, who played in a protective mask having sustained a broken nose in Palace's 3-2 win at Everton last time out, drilled goalwards from a cleverly worked 19th-minute set-piece.
His shot appeared to strike Nolan on the arm, but no penalty was given and referee Martin Atkinson was similarly unmoved after 32 minutes when Jedinak bundled Carroll to the floor.
In between those two incidents, Julian Speroni tipped over a rasping drive from Mohamed Diame and the Crystal Palace goalkeeper superbly denied Diame and Carroll from close range in quick succession as West Ham ended the half on the front foot.
Carroll pulled a 50th-minute shot wide of the near post having collected Stewart Downing's left-wing cross and the striker was then denied by Damien Delaney's last-ditch block.
Yet it was Palace who broke the deadlock from the spot in the 59th minute.
Having survived a penalty claim from Bolasie earlier in the half, Armero was not afforded another reprieve as he sent Jerome clumsily to ground and Jedinak dispatched the resulting penalty emphatically.
The visitors then swarmed forward with intent and wingers Bolasie and Jason Puncheon both had chances to extend Palace's advantage.
A second away goal proved elusive, while West Ham substitute Joe Cole twice blasted off target inside the final 10 minutes, but it was a satisfactory win for Tony Pulis and his team to conclude a week dominated by accusations of team-leaking ahead of their recent win at Cardiff City.