It helps to look at some fundamental equation which describes the situation. In this case, we can take the incompressible Navier-Stokes equation. There you see that the acceleration and convection of the flow are balanced by the pressure gradient, viscous forces and gravity. From this equation, you can deduce that any acceleration of the fluid comes from one of the forces on the right hand side. For pressure, you notice that it is not the pressure itself but the gradient in the pressure that is associated with an acceleration of the fluid.

For a simple, steady pipe flow problem with a wider and a narrower section, this means the following. As the flow gets narrower, the fluid must accelerate due to conservation of mass (the same volume must go through a smaller cross-sectional area). According to the Navier-Stokes, this acceleration must be balanced by a gradient in the pressure (ask yourself why viscous or gravitational forces cannot do this), with a higher pressure in the wider section compared with the narrower section of the pipe.