Alex Iwobi's first Everton goal helped seal a 4-2 win at third-tier side Lincoln in the League Cup second round on Wednesday as the top-flight club survived a scare, while Burnley were dumped out by Sunderland.
Everton, who have scored just one goal in three Premier League games this season and lost 2-0 at Aston Villa last week, fell behind after only 20 seconds before fighting back to lead 2-1.
Lincoln's Bruno Andrade scored a stunning 70th-minute equaliser, but Iwobi nodded in and Richarlison scored late on to send Marco Silva's men into round three.
"It was a difficult game. Credit to Lincoln. The important thing is we're through. We showed our fighting spirit," said Iwobi.
"I'm enjoying this new chapter of my life. Every competition we take seriously."
Close-season arrivals Moise Kean, Djibril Sidibe, Iwobi and Fabian Delph all made their full debuts for Everton, but the visitors suffered a disastrous start at Sincil Bank as Harry Anderson slotted into the net after a burst forward from Jack Payne.
Everton quickly began to dominate proceedings, but had to wait until the 36th minute to equalise as Lucas Digne curled a magnificent 30-yard free-kick into the top corner.
Teenage striker Kean, signed from Juventus earlier this month for 27.5 million euros ($31m, £25m), almost scored his first goal in English football shortly before half-time but saw his powerful effort crash against the near post.
But Everton did take the lead in the 59th minute as Morgan Schneiderlin was tripped in the box, with Gylfi Sigurdsson smashing the resulting penalty into the roof of the net.
A scuffle ensued inside the goal as Lincoln players tried to take the ball from Sigurdsson, with Everton goalkeeper Jordan Pickford running the length of the pitch to get involved and earning a booking for his troubles.
The League One outfit did not lie down, though, and levelled with 20 minutes to play thanks to a thunderous volley from Portuguese winger Andrade.
But Iwobi got off the mark for his new club after joining from Arsenal on deadline day, heading in from close range in the 81st minute just as he was set to be replaced by Theo Walcott.
And Richarlison also scored his first goal of the campaign with a powerful header from Digne's excellent cross.
"It's an important win for the season," said French left-back Digne.
"When I can score it's good for the team. We have to defend better but we're happy with the win."
Burnley bow out
Elsewhere, Premier League side Burnley were dumped out at the first hurdle after letting an early lead slip in a 3-1 home loss to third-tier outfit Sunderland.
Jay Rodriguez scored his first Clarets goal since returning from West Brom, but Sunderland sealed a shock win thanks to goals from Will Grigg, Tom Flanagan and George Dobson.
Kasper Schmeichel saved two spot-kicks as Leicester knocked out Newcastle after a 1-1 draw, with Yoshinori Muto and James Maddison scoring the goals in normal time of an all-Premier League tie at St James' Park.
Bournemouth narrowly avoided an early exit, beating Forest Green on penalties after a 0-0 draw as the fourth-division team missed all three of their spot-kicks.