Xherdan Shaqiri put Liverpool head on nine minutes, while Takumi Minamino added the second goal.
Curtis Jones fired home a brace with Minamino, Marko Grujic and Divock Origi scoring the other goals for Jurgen Klopp’s side.
Tayo Edun and Lewis Montsma were on target for Lincoln.
At the Etihad Stadium, Manchester City defeated Bournemouth 2-1 to also book a place in the next round.
Former Flying Eagles midfielder Nnamdi Oforboh featured for Bournemouth in the game.
Aston Villa are also through to the fourth round following a 3-0 win against Aston Villa.
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