The game's first opportunity should have led to Thistle's opener after 11 minutes. Jukka Santala found himself one on one against the goalkeeper but hesitated and his weak shot was easily cleared.
The large travelling support had to wait a full 15 minutes before they had something to cheer when Chris Templeman fired an 18-yard shot which was well saved.
A minute later Santala had another chance to open the scoring, but his 18-yard volley flashed the wrong side of the bar.
The high tempo of a pulsating first half was kept up, and Morton had the best chance after 19 minutes when a Walker snap-shot could only be parried by Colin Stewart in the Thistle goal, but James McAlistair's follow up was well saved.
Both teams continued to look for the opener and it fell to the home side after 29 minutes when Graham Dorrans combined with Ricky Gillies and Dorrans' 18-yard shot appeared goal-bound until it was blocked by the Morton defence. The ball fell kindly for Santala, who this time made no mistake and fired the ball high into the roof of the net from about six yards.
As half-time approached Thistle doubled their lead when a Scott McCulloch thunderbolt from a free-kick gave the Morton keeper no chance as it fizzed into the net.
The second half produced few chances for either side as Thistle were able to repel any chance that Morton had of taking anything from this game.
The final whistle was greeted with delight from the Thistle supporters and derision from the Morton fans, who saw their side slip up in the race for the championship.