Boro, eighth in the second-tier Championship, followed up their fourth-round win over Manchester United by defeating Tottenham Hotspur 1-0 on Tuesday.
Manchester City visit Southampton in an all top-flight encounter, while Liverpool are assured of Championship opposition after being drawn against the winner of Nottingham Forest and Huddersfield, who play on Monday.
Crystal Palace will face Everton in another all-Premier League Last 8 tie.
The quarter-finals will be played over the weekend of March 18-21.
Draw for quarter-finals of FA Cup:
Crystal Palace v Everton
Nottingham Forest or Huddersfield v Liverpool
Middlesbrough v Chelsea
Southampton v Manchester City