They both advanced to the last 16 on Tuesday but took different routes.
Chelsea came from two goals behind at Premier League champion Leicester, taking an end-to-end game to extra time at 2-2 thanks to goals from defenders Gary Cahill and Cesar Azpilicueta before Cesc Fabregas' double clinched a 4-2 win.
Liverpool had it much easier, dispatching second-tier Derby in a 3-0 victory with Ragnar Klavan, Philippe Coutinho and Divock Origi scoring the goals.
Chelsea and Liverpool both rested key players but still fielded strong lineups in a bid to stay alive in one of three competitions they'll enter this season.
More From This Section
Leicester has much bigger priorities this season. Aside from attempting to retain its Premier League title, Claudio Ranieri's team are in the Champions League for the first time and likely will not be too concerned about exiting a competition of the least importance.
Japan forward Shinji Okazaki took advantage of defensive errors to score twice for Leicester at King Power Stadium, but the hosts were undermanned in extra time after Marcin Wasilewski collected his second yellow card in the 89th for elbowing Diego Costa in the face.
Everton failed to transfer its strong Premier League form to the League Cup by losing to second-tier Norwich, which played almost a complete reserve team.
Steven Naismith, a former Everton player, and Norwich double their lead in spectacular style as one of the Murphy twins, Josh, spins with his back to goal maybe 18 yards out, unleashes a left-footed strike that whizzes in off the underside of the bar!Josh Murphy were the scorers.
Lucas Perez, a recent signing from Spanish side Deportivo La Coruna, scored twice in the second half for his first goals for Arsenal.
Bournemouth joined Everton in exiting the competition, meaning six Premier League teams have been eliminated so far. Danish striker Simon Makienok scored a hat trick for second-tier Preston, with his second goal coming in the 85th minute to take the game into extra time and his third clinching the win in the 111th minute.
Fresh off a heavy defeat at Manchester City, Bournemouth played an entire second-string team.