David Clarkson’s calm finish after 41 minutes gave the hosts the advantage leading into the break all for Graham to convert from the spot eight minutes into second half to ensure the spoils were shared.
After picking up early cards for consistent fouling, Dundee pair Gary Harkins and Clarkson initiated a period of sustained pressure for the hosts.
However Martin Boyle, Paul McGowan and James McPake, along with the cautioned duo, all saw efforts saved by the impressive Alan Mannus in the St Johnstone goal, as Dundee failed to capitalise on their ascendancy.
Their early dominance eventually paid off just four minutes before half time when Clarkson latched onto Greg Stewart’s pass and, after controlling, guided the ball past Mannus to give Dundee a well-deserved lead.
The goal ensured Clarkson extended his fine record of scoring in six consecutive games for the Dens since his switch from Bristol Rovers in September.
Their lead, however, lasted just eight minutes into the second period after referee Bobby Madden adjudged McPake to have fouled Michael O’Halloran in the area and pointed to the spot. Brian Graham stepped up and steered the ball low into the bottom left-hand corner to level the game.
Buoyed by their equaliser, the Saints pushed for a would have been a deciding goal but Graham’s drilled shot from distance was well held by Dundee goalkeeper Bain.
Missed stoppage-time efforts from Graham, again, and McPake ensured the game would end level as Dundee moved up to sixth in the Premiership table, while St Johnstone drop a place to ninth in the table, despite their point.