Bell held Matt Lockwood’s 64th-minute effort after Steven Milne levelled for Dundee 20 seconds after the restart.
Manuel Pascali headed Kilmarnock into the lead in the 32nd minute after the visitors had earned a reprieve when Milne had a goal very belatedly disallowed, although he was well offside.
Kilmarnock pushed strongly for a winner but they needed two more excellent saves from Bell to secure a replay.
Dundee produced some impressive passing moves throughout and were the first of the teams to settle into a rhythm as they created a series of early chances.
Right-back Gary Irvine burst through after an exchange of passes but could not get any power in his shot and Bell saved.
Bell then held Ross Chisholm’s volley after good skill from the former Hibernian midfielder and a return ball from Gavin Rae.
Rae soon had a better chance when Momo Sissoko failed to clear Ryan Conroy’s cross on the edge of the six-yard box but the former Scotland international’s shot was deflected wide.
Kilmarnock, with James Dayton and Rory McKeown back following injury, created their first chance when Paul Heffernan cut the ball back for Dean Shiels, but the loan player could not adjust his body properly and shot wide.
Dundee thought they had the lead in the 15th minute when Milne stabbed the ball past Bell from inside the box as the visitors waited in vain for an offside flag.
Milne was several yards offside when Chisholm flicked on Conroy’s initial pass and the Kilmarnock players surrounded referee Stevie O’Reilly, who had pointed to the centre spot.
O’Reilly changed his mind after talking to assistant Charlie Smith, who presumably missed the flick-on. The controversy seemed to rob Dundee of their momentum but the opening goal still came against the run of play.
The home defence could not match Pascali’s near-post run and the Italian got to Shiels’ corner before goalkeeper Rab Douglas to head home.
Dundee almost gifted the Ayrshire side another goal with some loose passing inside their own half but Shiels shot just wide.
Kilmarnock improved in possession but Rae had a decent chance just before the break when he got in behind the visitors, only to shoot over from 20 yards.
Rae immediately made amends after the restart as he drove forward to instigate the equaliser.
He laid the ball out to Conroy whose low ball across the six-yard box was steered home by Milne.
Kilmarnock tried to hit back and Douglas produced a stunning save to tip over Garry Hay’s powerful 20-yard strike before Pascali headed the resulting corner just wide.
Dundee had an ideal chance to move in front when Sissoko was adjudged to have pushed Conroy as the pair went to meet Jamie McCluskey’s cross.
But Lockwood missed his third penalty in a row as his weak low effort allowed Bell to dive to his right and hold.
Pascali soon headed wide from another corner by Shiels, who was proving more influential after dropping deeper.
The Northern Irishman opened up the Dundee defence with a pass to substitute William Gros and Heffernan appeared to be pushed by Lockwood as he attacked the winger’s cross, but no penalty was given.
Douglas saved Liam Kelly’s header following another Shiels corner before Bell made two excellent stops in quick succession.
The Scotland international came off his line to block from Milne before recovering to dive low to push away Conroy’s header.
Heffernan had a close-range shot blocked as Killie failed to avert a Rugby Park replay a week on Tuesday.