You mention the expansion of the universe, so I'm assuming your question is in the context of General Relativity.
The first thing to make clear is that the expansion of the universe is an expansion in spacetime and not just an expansion in space. We see it as an expansion in space because we are comoving observers, but this is an accident of the co-ordinate system we're using rather than a fundamental split between space and time.
In fact, in General Relativity the difference between the spatial co-ordinates and the time co-ordinate is not fundamental, and different observers will find their views of what constitutes time and what constitutes space will differ. There is not really a flow of time in GR any more than there is a flow of space.
So the expansion of space does not explain the one way direction of time. Over the decades (millenia if you include St Augustine!) there has been much discussion over why there is a directional flow of time but no-one has reached any satisafctory conclusions. If you're interested I recommend Googling for Sean Carroll's articles on the subject as he's particularly interested in this area. See for example this video for a popular science level introduction to the subject.