Difference between voltaic cell in chemistry and battery in physics
I think you've confused the flow of electrons with the flow of current. Before it was discovered that electricity is the result of the flow of negatively charged electrons it was proposed that electricity result from the flow of a positive charge. We called this current and this convention has remained to this day. 

So current is effectively the flow of positive charge not the flow of electrons, which are negatively charged. So current flows from the cathode to the anode, but electrons flow from the anode to the cathode. This is true in both physics and chemistry.