As I see it, gravity is the driving force in a siphon and air pressure only provides a supporting role. If you look a link chain over a pulley and rail up the pulley so that both ends of the chain are off of the ground, the chain will spill off of the pulley if the chain isn't perfectly balanced on it - that is, if the hanging ends of the chain aren't the same length. That's gravity at work.
Think of the water in your siphon tube as being that chain. The weight of the water in the down-flowing part of the tube weighs more than the water in the up-flowing part, so it pulls the shorter tube full of water up and over the "pulley".
Here's where it gets tricky. Why doesn't the down-flowing water separate from the up-flowing water at the top highest point in the tube? Consider the case where it does separate. What fills the gap in between? Air? No - there's no way for air to get in there because the tube doesn't let air in. A vacuum? Can't be a vacuum- the air pressure on the water above would force the water to fill that void. Another way of putting it is that the vacuum would suck the water up the tube just like someone drinking water through a straw.
Maybe air could bubble up from the bottom of the tube and fill that gap. Here I would say that the cohesion properties of water come into play. The air can't break through the water because the water won't move aside for it. Water cohesion is that force that causes water to stick together and form droplets.
Some questions to ponder to verify what I wrote are these:
1. Can a siphon work in zero gravity? I say "no" - there's not sense of up and down to begin with so water doesn't "flow." This is why astronauts drink their Tang from squeeze pouches. Gravity is required.
2. Can a siphon work in gravity but in a vacuum? This one might be debatable. The water *could* separate at the top of the tube, allowing a vacuum to form and resulting in the tube draining through both ends. I have heard the argument that water cohesion would prevent this. Maybe so, but there are other liquids without the same about of cohesion that would not siphon in a vacuum.
3. What if the tube was much larger in diameter on the uphill part than the downhill part, so that the weight of the water on the uphill part was greater than that on the downhill part? Would the siphon still work?