Sampdoria were leading twice in London, but fitness levels were decisive as West Ham won 3-2.
The Blucerchiati visited the Boleyn Ground for the inaugural Marathonbet Cup, which saw ex-Lazio striker Mauro Zarate start for the Hammers.
Samp took the lead in London after 16 minutes, as Stefano Okaka got in front of Collins to slot home an Eder through ball.
Nenad Krsticic and Eder wasted opportunities, but in first half stoppages Lorenzo De Silvestri brought down Potts for a penalty that Mark Noble converted.
Manolo Gabbiadini should’ve made it 2-1 at the dawn of the second half, but was put off by a last-ditch Potts sliding challenge, then two efforts were charged down from Okaka and Gabbiadini.
Sampdoria took the lead for the second time that afternoon, as Eder got between centre-backs to run on to a Roberto Soriano pass and beat the goalkeeper.
Ex-Catania striker Gonzalo Bergessio was given his Samp debut, but substitute Mo Diame got the Hammers back on level terms with a stunning goal, blazing past both Fabrizio Cacciatore and Andrea Costa.
They almost turned it around completely, as their season begins two weeks earlier than the Italian one and it began to show in their fitness levels, but the bar kept out Downing’s header.
Right at the death 17-year-old Burke was left totally unmarked to side-foot in a Lee pass from six yards. By this stage the Blucerchiati had visibly run out of steam.
West Ham 3-2 Sampdoria
Scorers: Okaka 16 (S), Noble pen 45 (W), Eder 61 (S), Diame 77 (W), Burke 90 (W)
West Ham United: Adrian, Tomkins, Collins (Burke 76), Reid, Cresswell (Potts 33), Kouyate (Diame 57), Noble (Poyet 85), Nolan, Downing, Zarate (Vaz Te 46), Cole (Lee 75)
Sampdoria: Da Costa; De Silvestri, Cacciatore, Gastaldello (Costa 72), Salamon; Palombo (Campana 82), Krsticic (Obiang 46), Soriano (Fedato 82); Gabbiadini (Rizzo 82), Okaka (Bergessio 67), Eder (Sansone 67)