The fifth-tier club became the first non-league team in 103 years to reach the last eight with the biggest shock of the competition so far on Saturday.
Holders Manchester United play Chelsea, and Millwall, who beat Premier League champions Leicester, play Tottenham.
Middlesbrough face Manchester City or Huddersfield, who drew 0-0 on Saturday.
Sutton, who are the lowest-ranked team still left in the cup, play at home to the Gunners in a 19:55 GMT kick-off on Monday that will be live on BBC One.
The replay between Manchester City and Huddersfield is provisionally set for Tuesday, 28 February at Etihad Stadium.
Full FA Cup quarter-final draw:
Chelsea v Manchester United
Middlesbrough v Huddersfield/Manchester City
Tottenham v Millwall
Sutton/Arsenal v Lincoln