Chelsea beat AFC Wimbledon and Everton eventually overcame Doncaster Rovers but both Premier League sides survived scares, coming from behind to claim 2-1 wins over their fourth-tier opponents in the League Cup second round on Wednesday.
Premier League Sheffield United, however, were sent packing as they lost to third-tier Lincoln City on penalties after a goalless draw.
Wimbledon took a shock lead from the penalty spot at Stamford Bridge in the 19th minute through James Tilley after Chelsea keeper Robert Sanchez brought down Harry Pell inside the box.
Chelsea continued to enjoy the lion's share of possession but failed to make it count until they were awarded a penalty for a foul on Noni Madueke, which the England under-21 forward converted in first-half stoppage time.
Substitute Enzo Fernandez grabbed the winner for the hosts in the 72nd minute with his first goal for the club since he joined from Benfica in a British record deal in February.
Not having qualified for Europe last season, Chelsea entered the competition in the second round for the first time since 2016, after finishing 12th in the Premier League.
Doncaster got off to a lively start against Everton as the bottom-placed League Two side took the lead against the bottom team in the Premier League.
The hosts gave Everton manager Sean Dyche a lot to think about at half time after they took the lead just before the breakthrough Joe Ironside's header following a corner.
Replays showed the goal appeared to be offside, but it stood as there was no VAR in operation.
Doncaster could have doubled their lead just before half-time but Jordan Pickford denied Tommy Rowe and another attempt from Ironside.
Everton, who have lost their three opening Premier League games without scoring a goal, faced a flurry of attempts from their lowly rivals before they equalised through new arrival Beto in the 73rd minute.
The 25-year-old, a half time substitute, sent a low shot past Doncaster keeper Ian Lawlor less than 30 minutes into his debut after signing for Everton on Tuesday and he could have sealed the win in the 82nd when he hit the post with a powerful header.
But it was forward Arnaut Danjuma who buried Doncaster's hopes two minutes from full time with an individual effort.
"Relief is the word we are looking for, it was a hard-fought game, I can't say we are satisfied," Danjuma told Sky Sports.
"We should have killed the game earlier."
In an all-Premier League duel, Burnley forward Zeki Amdouni scored in the 90th minute to knock out Nottingham Forest 1-0 in a drab encounter that saw no shots on target for over an hour.
Newcastle United will face Manchester City in the third round, after the draw was made on Wednesday, while Chelsea will take on Brighton.