Full-time from The Den
Smith, who was playing non-League football himself for Folkestone Invicta last season, produced clinical finishes on 17, 20 and 47 minutes to ensure that there was no giant-killing act by National League outfit Braintree as he took his goals tally to six in six starts for The Lions.
The travelling fans were still celebrating when Millwall hauled themselves level within 60 seconds of going behind. Lee Gregory did well to hold off his marker, spun on the edge of the penalty area and threaded the ball to Smith, who made no mistake from close range.
The Lions got their noses in front three minutes later with Smith's second goal of the game, this time a crisp half-volley after Gregory chalked up assist No.2 by steering Shane Ferguson's corner into his path at the back post.
Harris' men looked to be in control when Ferguson drilled home a third beyond the reach of keeper Sam Beasant after Gregory's sublime pass had picked him out on the left. Ferguson shimmied his way past a couple of defenders and made no mistake with a crisp finish into the far corner with 23 minutes on the clock.
Braintree, at this stage, were looking vulnerable every time Millwall progressed beyond the halfway line and decided that attack was going to be their best form of defence as they began to ask questions of their own.
A neat passage of play between Sean Clohessy and Simeon Akinola offered further encouragement for the visitors, and they were right back in the mix 11 minutes before the break when Jack Midson poked the ball home from a central position after a swift break down the right.
Smith and David Worrall both went close as The Lions looked to restore their two-goal cushion, but it was down to the athleticism of Archer that Braintree didn't go in level, as the keeper somehow kept out Monty Patterson's goal-bound effort.
Smith was a whisker away from adding his fourth - and Millwall's fifth- with a downward header from Ferguson's hanging cross, but the ball dropped inches wide of the upright.
But as the game began to open up, back came Braintree and Akinola hit a thunderous 18-yarder just over as the non-League side refused to give up on their dream of producing an upset.
Ian Gayle became the first player to be treated for cramp with 21 minutes remaining and he was ultimately replaced as the game entered the final 10 minutes after requiring treatment again.
Lions sub Aiden O'Brien, a 78th-minute replacement for hat-trick hero Smith, hit a low shot that was blocked and then fellow sub Fred Onyedinma nearly made an instant impact with a fierce drive that was diverted by the boot of a Braintree defender as The Lions searched for a fifth goal to really put the result beyond reach.
But it was Millwall who had the final with the last kick of the game - a deft flicked finish by O'Brien from barely four yards after fellow sub Onyedinma and Worrall had combined to set him up with a gilt-edged opportunity.
Booking: Webster (34')
Scorers: Smith (17, 20, 47'), Ferguson (24'), O'Brien (90+4')
Braintree Town: Beasant, Okimo, Lee, Gayle (Barnard, 84'), Midson, Isaac (Maybanks, 88'), Akinola, Jules, Cheek, Patterson (Matthews, 62'), Clohessy. Subs not used: Strizovic, Williams, Hall-Johnson, Corne.
Bookings: Isaac (58'), Gayle (60')
Scorers: Cheek (16'), Midson (34')
Att: 3,345 (away 315)