Goals in each half from Danny Graham and Graham Kavanagh earned Carlisle a second round home tie with Crewe.
In the second staging of the replay following floodlight failure last week, non-league Grays missed early chances.
Graham punished them in the 26th minute by turning in his 11th goal of the season after a cross from Jeff Smith.
Graham later crossed for Kavanagh to bundle into the net, while Marc Bridge-Wilkinson's penalty rebound was ruled out for handball by Fabian Wilnis.
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Subs Not Used: Arnold, Elliott, Hickie, Davis, Townsend.
Booked: Stuart.
Carlisle: Williams, Raven, Livesey, Keogh, Horwood, Taylor,Bridge-Wilkinson (Dobie 79), Kavanagh (Lumsdon 86), Jeff Smith, Graham, Madine (Welsh 79).
Subs Not Used: Howarth, Bridges, Blake, Aldred.
Booked: Kavanagh.
Goals: Graham 27, Kavanagh 52.
Att: 1,217
Ref: David Phillips (West Sussex).
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