The Pittodrie club took a first-half lead through Adam Rooney but Dundee levelled after 69 minutes when Stephen McGinn netted his first goal for the club.
Dundee deserved their point while the Dons must feel frustrated at not closing the gap on Celtic at the top of the table.
Dundee had claims for a penalty after 22 minutes when Simon Ferry's ball over the top put Paul McGowan free.
Jamie Langfield advanced off his line and the Aberdeen keeper's challenge saw Dundee midfielder Ferry fall inside the box. The hosts felt they had been denied a stonewall spot-kick but referee Craig Thomson was having none of it.
That gave the home team a boost and shortly after, Alex Harris' curled corner picked out German defender Thomas Konrad, only for him to head straight at Langfield when he ought to have done better.
Dundee were buoyed and soon after they threatened again. A move involving Ferry and Harris saw Stephen McGinn set free and, with David Clarkson making a decoy run, the midfielder drove a 20-yard effort just inches wide of the post.
But the Dons took the lead in style against the run of play in 36 minutes. Andy Driver picked up possession and slipped a pass into the feet of Rooney who, from 10 yards, sent a shot into the top corner past helpless Bain for his fourth goal in their last four outings.
And four minutes before the break, Dons were denied what looked like a clear spot-kick.
Niall McGinn's flicked shot came off Willie Dyer and the visitors immediately claimed it was off his hand, but referee Thomson insisted it came off his head so awarded a corner, much to the Dons' disgust.
That could have given Aberdeen the cushion they were looking for going into the break but they went in search of a second at the start of the second-half.
Three minutes after the restart Shay Logan's cross found Rooney who got slightest of touches, only for Bain to defy him with an instinctive close-range save.
Dundee fought back and they should have been level on the hour mark. Stephen McGinn nodded the ball into the path of former Motherwell man Clarkson, but with the goal at his mercy, he hooked his effort wide.
However, the home team levelled nine minutes later.
Harris clipped a cross into the box where McGinn cutely it in at the near post, past Langfield.
Dundee introduced young striker Craig Wighton in the closing stages in a bid to force a winner but it wasn't to be.