Russian Time driver Evans, who is mentored by former Red Bull F1 star Mark Webber, emerged from the pits ahead of Palmer after both had taken their mandatory stops. The Brit briefly threatened, but an off-track moment allowed Evans to ease away and win by almost five second.
Raffaele Marciello had started from pole and led early on, but his gearbox gave out and ended what looked set to be his maiden victory, paving the way for Palmer and Evans to battle for the win.
McLaren junior driver Stoffel Vandoorne was third with Julian Leal snatching P4 from Stefano Coletti just a few turns from the chequered flag. Next up were Johnny Cecotto Jr. and Felipe Nasr, while Stephane Richelmi took eighth and the reverse grid pole. Behind him Marco Sorensen impressed on his series debut by scoring points in ninth.
The Sprint Race saw Nasr triumph for a third time in 2014, keeping him second to Palmer - albeit some distance behind.
Nasr lined up second on the reversed grid, but a poor getaway from poleman Richelmi allowed the Brazilian to snatch the lead immediately.
He was rarely challenged thereafter. Coletti was quickly into P2 and briefly threatened to launch an attack, but when the Monegasque closed him down Nasr responded in kind to rebuild a comfortable gap. The Racing Engineering driver later reported losing power and was relieved to hang on to second, with Cecotto Jr. completing the podium.
Palmer was a solid fourth after a number of incisive passes, while Leal was fifth. Richelmi dropped all the way to sixth, with Evans seventh and newcomer Sorensen again in the points in eighth.
The Silverstone weekend's results see Palmer extend his advantage over Nasr to 38 points despite not taking a win for the second successive round. Cecotto is third and Evans leaps to fourth, level on points with Coletti. The series returns to action in two weeks' time, supporting the German Grand Prix at Hockenheim.