This may not be exactly what you are looking for, but I think it gets sort of close and it’s certainly interesting. If your planet was a moon and orbited closely around a much larger body then tidal effects could explain all of your water flowing in one direction.
Tides on Earth are quite complex. They are a result of the gravitational influence of the Sun and the Moon on the Earth. To put it simply, when one of these bodies is directly overhead, the gravity a person (or a body of water) feels is slightly reduced because the overhead mass exerts a slight pull towards it. This results in a high tide in our oceans because they are large fluid bodies that can react to these subtle forces very noticeably. There is also a high tide on the opposite side of the planet at the same time as a result of the rotational mechanics. This is not intended to be anything close to a good explanation for how tides work, but it will suffice for my suggestion.
If your planet is orbiting closely around a large planet it will have much larger tidal effects than we do on Earth. If your planet spins toward the East, like the Earth does, then the side of your planet facing the other large body will move West, much like the side of the Earth facing the Sun moves West. With exaggerated tides this could cause much of the water on your planet to “flow” West, to continue facing the large body your planet is orbiting. Essentially the large body would exert a pull on the majority of the water on your planet in a Westward direction.
What affects millions of years of these tides would have on the formation of the rivers and lakes and oceans I really can’t say. But it seems possible to me that it would result in the majority of the planet’s major rivers flowing West and carving out large valleys that have large diurnal tides.