The early chances came West Brom's way when Rodriguez hit a half-volley from inside the middle of the box, but he clipped it slightly and the ball ended up getting hoofed away by Daniel Amartey.
The West Brom new boy got on the scoresheet three minutes later after latching on to Jake Livermore's pass on the edge of the box and curling the ball into the back of the net.
Leicester responded later in the half when Mahrez, who submitted a transfer request at the end of last season, made the most of some poor play from West Brom.
Phillips's chipped ball towards Claudio Yacob missed its intended target, which allowed Mahrez to charge forward with the ball, slicing through defenders and slotting home from inside the box.
Were it not for some poor control from Islam Slimani, Leicester could have been two goals to the good before the break, and Mahrez had another opportunity when he set himself up to shoot from a rebounded effort, but Ben Foster made the save.
Both managers rung the changes for the second half, and it was West Brom who came out firing as Livermore played the ball through to James McClean, but his powerful effort went wide.
Nacer Chadli tried to nudge the Baggies ahead from a free kick, but Leicester's goalkeeper got a hand to it and prevented the ball from nestling into the net.
With the heat causing the teams to stop for regular drinks breaks, it was never going to be a ferocious encounter, but the two Premier League sides failed to find a winner in normal time, which resulted in a penalty shootout.
Spot kicks were to be taken in an unorthodox "ABBA" style, whereby West Brom would line up first to take one, then Leicester would take two, then the Baggies would take another two and so on.
There were 13 penalties in all, with 19-year-old West Brom midfielder Sam Field the only individual to miss his effort, while the winning spot kick was converted by Albrighton.