Managed to beat it too this morning, after spending yesterday evening devising a strategy through trial and error (and yes, I got a night's sleep in between).
Basically, keep the high numbers along one side. For me, that was the bottom side. Also try to always shuffle tiles in the same direction. You might also want to push the "biggest" tiles into a corner, though that make them a lot harder to reach. Having your highest numbered tile next to the corner is a lot better. That way, it can be reached from several sides by shuffling. Since you shuffle left and right a lot, small tiles will generate next to the edges of the board all the time, and make any corner tile inacessible from above.
Another key point of the strategy: Guard the bottom row. Do not allow any 2 or 4 tiles to build up on the same row as your highest numbered tiles. They will fill up the board unnecessarily, and be almost impossible to remove. Look at the "game over" screencaps posted on this page. All but one of them have fallen into that trap.
Other than that, Tetris strategy it is. Always try to splice tiles vertically instead of horizontally, to push the biggest numbers as far down as you can. When splicing horizontally, plan ahead so the right tiles appear above another (so you can splice vertically). And never, ever, use the up key. That will generate small-numbered tiles on the bottom row, and if a "barrier" of big tiles are between them and the rest of the board, you'll never get rid of them.
