A much-changed Toffees outfit began slowly against their Championship opponents, who employed a pressing game in a bid to unnerve the relatively inexperienced hosts.
Marco Silva's charges did manage to make headway into the opposition half on eight minutes, a good move culminating in Lucas Digne pulling the ball back to Oumar Niasse, but the latter hesitated and nothing came of it.
From then on, the Merseysiders were on the up but spurned multiple opportunities to extend their lead, with Sigurdsson guilty of inexplicably scooping a point-blank effort over the bar.
The first half drew to a close with Everton just the one goal in front and the hosts set about trying to get more goals right from the start of the second period.
Dominic Calvert-Lewin eventually succeeded in doubling his side's advantage just after the hour mark, Digne playing a cross to the back post for the 21-year-old to nod home from several yards out.
The Millers were left with a mountain to climb and nearly conceded another on 66 minutes, but Niasse missed the target with a header from Jonjoe Kenny's pinpoint cross.
Four minutes from time, Will Vaulks managed to halve the deficit with a header at the near post from a corner, seemingly leading to a nervy finish for Everton.
However, Calvert-Lewin made sure of progress just moments later with a smart curling finish to finally break Rotherham's resistance.
EVERTON (4-4-2): Stekelenburg; Kenny, Holgate, Zouma, Digne; Ramirez (Schneiderlin 86'), Dowell, Davies, Sigurdsson (Walcott 66'); Calvert-Lewin, Niasse (Tosun 73')
ROTHERHAM UNITED (4-4-1-1): Price; Vyner, Robertson, Raggett, Ajayi; Forde, Palmer, Ball (Vassell 61'), Wiles (Vaulks 46'); Newell (Taylor 72'), Proctor