Two goals from Dominic Calvert-Lewin and Gylfi Sigurdsson's first of the season ensured an under-strength Everton side avoided a Carabao Cup upset with a 3-1 victory over Sky Bet Championship side Rotherham.
Iceland international Sigurdsson, who netted six times last season following his £45m club record move from Swansea, scored from close range after 28 minutes, followed in the second half by Calvert-Lewin heading his first since February.
Substitute Will Vaulks nodded in an 86th-minute corner, but Calvert-Lewin's second moments later snuffed out any hopes of a late comeback.
Sigurdsson, along with Tom Davies - the academy graduate becoming Everton's youngest starting captain aged 20 years and 60 days - and Mason Holgate were the only three retained from Saturday's draw at Bournemouth.
On-loan Chelsea defender Kurt Zouma was relatively untroubled on his first start and is set to keep his place for this weekend's Premier League visit of Huddersfield with Phil Jagielka, suspended for this game, ruled out for three
weeks with a knee cartilage problem.
A setback in Andre Gomes' recovery means the on-loan Barcelona midfielder will be out for a further three weeks, winger Bernard was missing with a muscle injury, Yerry Mina is out until after the international break with a foot problem while Richarlison has started a three-match suspension.
Added to that were the absences of midfielder Idrissa Gana Gueye (knee) and RB Leipzig target Ademola Lookman (groin).
It meant Silva named an-all new front three of Oumar Niasse, Calvert-Lewin and Sandro Ramirez - a trio who had managed just two goals in their last 38 appearances combined.
Spaniard Sandro provided the low cross for Sigurdsson's goal, but he offered little else.
Calvert-Lewin's far-post header from Digne's cross on the hour in front of a crowd of 31,972, Everton's highest for a home League Cup second round tie since 1968, established an important two-goal cushion.
And when Vaulkes nodded in late on Everton went straight down the other end for Calvert-Lewin to curl home his second.
VAR's first appearance at Goodison Park also passed without incident.