Yes. You win when a 2048 tile appears, and you can keep playing afterward to chase a bigger tile and a higher score.
You win 2048 the moment a tile worth 2048 or more appears on the board. The game announces the win, and you can then keep playing to reach 4096 and beyond.
Winning is not the end
Reaching 2048 does not stop the game. Carry on with the same board, and the game only ends when you run out of moves (when 2048 is game over).
Getting there
You need eleven doublings from the first 2, so keep your biggest tile in a corner and merge toward it. The 2048 strategy guide explains the corner method, and the full rules list every rule in order.