The Millers blew Phil Brown's men away in the opening 45 minutes at the AESSEAL New York Stadium, with Joe Newell adding a fine goal to Moore's treble and Ryan Williams notching after the break.
Moore got the ball rolling in the 15th minute when he slotted home from Williams' pass and then 60 seconds later he added his second, tapping home Jamie Proctor's pinpoint cross.
Newell made it 3-0 after only 23 minutes with the best goal of the lot as he fired beautifully into the far corner and then Moore became the first man to score a competitive hat-trick at New York when he converted from 12 yards in first-half injury time.
Williams made it five just before the hour when another brilliant team move left him with the simplest of tap-ins.
The Millers threatened to equal their record home win at New York but David Ball was denied by a good save from Mark Oxley while Southend were denied a late consolation when Marc-Antoine Fortune hit the post with the goal gaping.