The match opened in dramatic fashion with a goal for both teams in the space of little over a minute.
The hosts were the first to strike six minutes in when midfielder Luke Murphy curled home a fine shot from 20 yards out.
Millwall were level in the eighth minute, however, when Scott Barron's cross from the left wing somehow beat Steve Collis in the Alex goal.
It took just eight minutes of the second half for Millwall to take the lead they warranted when Harris prodded home Martin's cross from three yards to break Teddy Sheringham's club scoring record with his 112th strike.
But Shaun Miller levelled in the 57th minute, bundling home in a crowded box from Billy Jones' cross.
The match had become more even but Alex could have few complaints as the visitors set up a fourth-round clash with Hull or Newcastle at the death, Zak Whitbread powering home an 86th-minute header from Marc Laird's cross.