The visitors started brightly and with only seconds gone Eddie Malone fired just over the crossbar and minutes later Derek Soutar saved to deny Alex Williams.
Simon Lynch put Dundee ahead on 25 minutes when he raced into the penalty box and lifted the ball over the on-rushing Paul Jarvie from 10 yards.
The visitors equalised on 37 minutes when Tom Brighton scored a great solo goal which saw him go past two Dundee defenders before firing low past Soutar.
Tom McManus then scored either side of the interval to give Dundee what looked like a comfortable two-goal advantage.
His first goal came after he got on the end of a Robert Mann free-kick to fire low past Jarvie on 42 minutes and on 51 minutes he scored from close range after Michael McGowan had crashed the ball off his own bar.
Ten minutes from time Clyde were handed a lifeline when Mann fouled Craig Bryson inside the box and Stephen O'Donnell scored from the spot.
With five minutes to go O'Donnell scored again to tie the game at 3-3 when a great run down the right by Brighton saw the young striker cross into the danger area where his team-mate was on hand to fire under Soutar from close range.