Preheat oven to 375°F. Line a heavy 10 x 15-inch jelly-roll pan with parchment paper.
Beat eggs on high speed until very thick and lemon-colored. Gradually beat in sugar. On low speed, beat in water and vanilla.
Whisk flour, cocoa, baking powder, and salt together. Gradually add to the egg mixture, beating just until smooth.
Spread batter evenly in the prepared pan and bake 12 to 15 minutes, until the top springs back when touched and a toothpick comes out clean.
Immediately loosen and lift the cake from the pan using the parchment. While hot, roll the cake from the short end using the parchment and let cool completely, at least 30 minutes.
Whip cream, vanilla, and powdered sugar to soft peaks.
Gently unroll the cooled cake, spread filling evenly over it, then roll it back up without the parchment. Wrap seam-side down in plastic wrap and refrigerate at least 1 hour.
Dust with powdered sugar before serving, if desired.