Dundee managed to take a surprise point off Rangers last week - but failed to find the net against St Johnstone.
Expectations were high for Ivano Bonetti's side after their result against the Scottish champions and although they had plenty of chances they could not find the killer touch to take the three points.
Fabian Caballero and Javier Artero came close but their chances were cancelled out by a menacing Graeme Jones who had several chances to put Saints ahead.
Caballero burst through the middle on a magnificent run in the opening seconds of the game before laying the ball off to Artero but he failed to connect properly in front of goal and the chance was wasted.
St Johnstone tried to hit back a minute later when Gary Bollan unleashed a powerful effort from 30 yards but the shot flew harmlessly past the post.
Marcello Marrocco faced a late fitness test before the game after taking a knock last week and he had to be replaced after just eight minutes by Shaun McSkimming.
Four minutes later Caballero should have grabbed the opener with a superb diagonal shot which had keeper Alan Main beaten but was just off target.
Dundee had been looking by far the busier team and Patrizio Billio had a go from just outside the 18-yard box. The shot deflected off Alan Kernaghan but worried Main enough to put the ball out for a corner in 22 minutes.
Saints started to push their way back into the game and should have been ahead after 25 minutes. McSkimming failed to clear a Paul Hartley free-kick properly but made amends when he had to clear the ball from Jones off the line.
A few minutes later and the home side were threatening again. Sylla's cross found Mick Dasovic whose pass picked out Jones who forced keeper Rab Douglas into a superb one-handed save.
On the half hour mark the home fans were in an uproar when Barry Smith and Jones were both caught up in a tangle in the box and the home side screamed for a penalty.
But their claims for a spot-kick were ignored by referee John Underhill.
Five minutes after the break Caballero found his name in the referee's book.
But Dundee remained undeterred and Gavin Rae almost snatched the opener but nodded just over from a corner six minutes later.
Just before the hour Jones had another great chance when he found himself clear on goal but the striker failed to realise how much time he had to place the shot and hastily blasted wide.
Promising young prospect Keigan Parker attempted to bring Saints back into the game but like so many other chances today his shot was off target.
A minute later Georgi Nemsadze tried to set up Artero but the Spaniard nodded weakly into the arms of Douglas.
In the last minutes of the game Jones almost set up Parker once again but the youngster failed to connect properly and the chance was lost.
In the dying seconds Parker looked certain to snatch the three points when he found himself with only the keeper to beat but Douglas dived low to block the shot.