After a goalless first half following a dramatic 3-3 draw at the Pirelli Stadium 11 days ago, Palmer fired League Two Burton in front with a smart free-kick in the 65th minute.
And no sooner had Conference North Altrincham settled, than Zola had doubled the Brewers advantage to set up a second-round clash at his former employers Crewe.
Altrincham had the first chance of the match after just eight minutes, however, striker Damian Reeves could only scoop over a smart right-wing cross from Scott Leather.
Burton responded in the 24th minute through Zola, whose late equaliser salvaged a replay for the Brewers, heading against the bar following a Palmer free-kick.
Altrincham striker James Lawrie saw a drive deflected wide after the break before Palmer broke the deadlock with a free-kick after Simon Richman fouled Jacques Maghoma.
Ten minutes later the visitors clinched victory as Maghoma teed up Zola to finish left-footed from the edge of the six-yard box and extinguish any hopes of an Altrincham comeback.
The Robins did come agonisingly close to setting up a tense finale, though, when Nicky Clee played in Matt Doughty but his teasing ball across goal drifted narrowly wide.