Rovers had the better opportunities but the League One side, who have reached the quarter-finals three times in the last eight years, were unable to break the deadlock.
Accrington were the first to threaten as an early corner was cleared as far as Jimmy Ryan, who blazed over from just outside the box.
Stanley then had a fright as a fierce Ryan tackle from 40 yards out flashed narrowly wide of his own goal.
Tranmere kept the pressure on with lone striker Bas Savage flicking on a dangerous Charlie Barnett free-kick into the arms of Kenny Arthur.
Visiting keeper Danny Coyne was able to gather the rebound after spilling a 20-yard drive by Ryan, but Barnett then fashioned the game's first chance.
The Rovers midfielder slipped in Ian Moore but Arthur rushed out to make a block before tipping over Antony Kay's header from an Andy Taylor free-kick.
Gareth Edds then wasted the best chance yet, Arthur making a fine save after a neat flick by Moore found Edds 12 yards out, and on the brink of half-time John Miles sent a tame effort straight at Coyne.
Rovers right-back Ryan Shotton hit a 40-yard strike inches over shortly after the restart before Edds fired over on the hour mark after Taylor's cross fell into his path.
Tranmere made an attacking change with Chris Greenacre replacing Edds, and moments later he found Chris Shuker in the box, only for the Rovers winger to blaze over.
Savage twice went close late on, but the two sides had to settle for a replay on Tuesday, November 18.