To naifmbo, the answer to your query is: yes ! Water in a pipe can move from lower to higher pressures. It is mechanical energy that moves the fluid, not just pressure.
Mechanical energy in frictionless flow (remember Bernoulli's expression for incompressible fluids) has three components: kinetic energy per unit mass (V2/2), potential energy per unit mass (Zg), and flow work (pressure multiplied by volume (=1/[ρ]).
If we divide all these components by g, we get the dimensions of length or height. V2/2g is called velocity head, Z is called height, and p/([ρ]g) is called pressure head.
BTW, [ρ] is density; V is velocity; p is pressure; g is acceleration of gravity.
Thus water can flow from point 1 to point 2 against a higher pressure if the other components of mechanical energy in point 1 are sufficiently greater than those in point 2 so as to overcome the negative pressure difference.