Dominic Calvert-Lewin fired a first-half double as Everton won 2-0 at Sheffield Wednesday in the Carabao Cup to ease the pressure on boss Marco Silva.
Calvert-Lewin, who progressed through Sheffield United's academy, sank Wednesday with two clinical finishes inside the opening 10 minutes to effectively kill the tie.
Silva's side had slumped to back-to-back Premier League defeats at the weekend, losing 2-0 at home to Wednesday's city rivals United, and there were murmurings of discontent at Goodison Park.
The Portuguese made five changes, recalling Calvert-Lewin to the starting line-up and the Sheffield-born striker responded in style in front of the Hillsborough Kop end.
Wednesday boss Garry Monk made eight changes following Saturday's 1-1 home draw against Fulham and his side showed little of the cohesion needed to have caused an upset.
It could have been very different had Wednesday not spurned a golden chance to take a second-minute lead.
Sam Winnall found himself one-on-one with Jordan Pickford after the Everton defence had been caught cold, but the England goalkeeper smothered the striker's shot and Wednesday were soon made to pay.
France international Djibril Sidibe, yet to start a league game this season, clipped a lovely ball to the far post after racing into Wednesday's half and Calvert-Lewin took one touch before lashing home an emphatic finish into the roof of the net.
Everton quickly followed that up with another clinical counter. Bernard's neat pass set Alex Iwobi free down the right and his pin-point cross was tapped home by the unmarked Calvert-Lewin.
Wednesday wasted further first-half chances. Dominic Iorfa headed off target at the back post and Adam Reach fired straight at Pickford after the ball had fallen for him inside the box.
Everton showed less urgency in the second half, while the home side plugged away.
Winnall fired one effort over the crossbar and headed another wide before the impressive Sidibe shot wide after another foray forward.
Wednesday threatened through Reach's trademark long-range effort, but it failed to trouble Pickford, while Everton skipper Lucas Digne pulled his shot off target in the closing stages.