Liverpool will face Arsenal in the pick of the ties in next week's Carabao Cup fourth round.
The Reds will welcome the Gunners to Anfield twice in the space of a matter of days after they face them in the Premier League next Monday.
Jurgen Klopp's side strolled into the last-16 with a commanding 7-2 win over League One side Lincoln City at Sincil Bank, where Xherdan Shaqiri opened the scoring with a stunning free-kick after nine minutes.
Takumi Minamino measured in a fine shot to make it 2-0 shortly afterwards, with local lad Curtis Jones grabbing two more goals to make it 4-0 by half-time.
Japan international Minamino scored his second in the opening seconds of the second half, with Tayo Edun and Lewis Montsma scoring for the Imps either side of a Marko Grujic strike, with Divock Origi scoring the seventh.
Manchester City had to battle somewhat harder to shrug off Bournemouth 2-1 at the Etihad Stadium.
City youngster Liam Delap - son of Stoke City legend Rory - gave the hosts the lead, only for the Championship side to level before the break through Sam Surridge.
Phil Foden then grabbed the winner 15 minutes from time.
Pep Guardiola's men now face a trip to fellow Premier League outfit Burnley next week.
Elsewhere on Thursday, Aston Villa eased past Championship side Bristol City 3-0 at Ashton Gate thanks to goals from Ollie Watkins, Bertrand Traore and Anwar El-Ghazi.
Dean Smith's men will host Stoke City in the next round.
The fourth round draw in full:
- Liverpool vs Arsenal
- Burnley vs Manchester City
- Brentford vs Fulham
- Everton vs West Ham
- Aston Villa vs Stoke City
- Leyton Orient/Tottenham Hotspur vs Chelsea
- Newport County vs Newcastle United
- Brighton & Hove Albion vs Manchester United
