- 1,612
- 9
- 38
In the 34th minute Graham connected with a Kyle Turner corner but Laidlaw was down smartly to save.
County edged forward as often as they could but they appeared susceptible to losing a second, with Graham thundering a drive over from the edge of the box.
The former Ross County and Dundee United forward was much more accurate from another Turner corner, volleying in from 12 yards to send the Glasgow side into the break on a high.
Ben Purrington replaced George Harmon, who had been booked, for the start of the second half as the Dingwall side looked to keep themselves in the tie but within minutes Thistle's Kevin Holt had volleyed over the Staggies crossbar from eight yards.
Thistle then swarmed all over the visitors and should have added to their lead.
Graham had a goal-bound drive blocked for a corner by centre-back Jack Randall before Scott Tiffoney also saw his effort from inside the box hit County defender Connor Randall and then Laidlaw somehow kept out a close-range header from Graham.
County hinted at a threat occasionally and in the 64th minute Jags defender Aaron Muirhead brilliantly blocked a shot from Staggies forward Jordan White.
The Dingwall side pushed for a lifeline which never arrived, the home side finishing in control with substitute Danny Mullen cracking the bar with a powerful drive and they will surely travel to Dingwall on Sunday more in belief than hope.