Alexander was probably relieved that his bizarre attempt to occupy the home dug-out failed and his successor Davie Irons could only watch and suffer as a first-half double from summer signing Michael Higdon set Falkirk on their way.
The result cements Gretna's position as relegation favourites but they started strongly at their adopted home of Fir Park and could have gone ahead in the first ten minutes.
Falkirk defender Darren Barr made a crucial headed clearance as striker Colin McMenamin lurked unmarked at the back post and Allan Jenkins then wasted a glorious opportunity after his one-two with McMenamin had ripped Falkirk apart.
Uruguayan forward Fabian Yantorno latched onto the loose ball but his deflected effort was easily cleared off the line by Jack Ross.
Falkirk opened the scoring with a goal of real precision. Pedro Moutinho found Tom Scobbie on the left and his cross was nodded into the far corner of the net by Higdon.
The ever-dangerous McMenamin forced an excellent save from newly-signed keeper Tim Krul and Yantorno blazed the rebound over the top.
Moments later Steve Thomson rattled the crossbar from 22 yards after being given too much time to take aim and Higdon showed Yantorno how to gobble up rebounds with a clever finish with the outside of his foot.
Higdon had two chances to complete his hat-trick early in the second period but headed weakly at keeper Greg Fleming and then rippled the side-netting from close range.
Falkirk's third goal summed up the match with Yantorno once again lacking conviction in the penalty area before Moutinho rounded off a swift counter-attack with a curling right-foot drive high into the top corner.
Chances continued to come and go for the Gretna strikers and Falkirk completed their misery 16 minutes from time when Moutinho set up midfield general Russell Latapy who drilled a low right-foot shot past Fleming.
With six minutes left Danny Grainger's left-footed free-kick beat Krul but came back off the post - it just wasn't Gretna's day.