Sutton bounced back from being sent off for a wild lunge at Arsenal's Patrick Vieira on Sunday to score within 60 seconds of the restart after half-time. He latched on to Damien Johnson's pass and drilled the ball beyond Jason Kearton and inside the far post.
Rovers, with Tim Flowers in impressive form, then hung on to complete only their third win in 13 attempts this term as their Worthington Cup campaign stayed on the right track against the Railwaymen.
Sutton played with a broken nose - the legacy of a clash with Vieira before the red card incident at the weekend - and while this was hardly a vintage performance, he turned out to be the difference between the teams. In the early stages, Blackburn struggled to prise open the Crewe defence. With the exception of the St Vincent international Rodney Jack, though, the home side looked limited up front.
The reward for Blackburn, when the fourth-round draw was made last night, was a visit to Newcastle United. The holders, Chelsea, have been given a trip to Arsenal, the team they beat to reach Wembley last year, after knocking Aston Villa out last night.
Gianluca Vialli, whose hat-trick helped beat Villa 4-1 at Stamford Bridge, can look forward to a repeat of last year's semi-final next month.
The fourth-round draw also paired Leicester and Leeds at Filbert Street. That tie sees the home manager, Martin O'Neill, take on the side that spent much of this month trying to lure him to the Elland Road before appointing David O'Leary as manager this week. The former Leeds manager George Graham takes his Tottenham side to Liverpool. Manchester United have home advantage against Nottingham Forest in the other all-Premiership tie.
In the Scottish Premier League, John Spencer, the brother-in-law of Motherwell's new manager Billy Davies, hit the winner on his debut in a 1-0 home victory over his old club, Rangers.
The Scottish Premier League leaders were beaten by a 55th-minute strike by Spencer, who is on loan from Everton - where he has fallen out of favour with the former Ibrox manager, Walter Smith.