Chris Duggan picked out Paul Woods on the left of the box and the Spiders midfielder slotted the ball into the back of the net.
The hosts struck back as Gary Fraser's free-kick from just outside the 18-yard box floated into the top corner.
The away side had numerous chances to clinch a winner but the spoils were shared at Links Park.
Queen's Park extended their unbeaten run to five matches but had to settle for a draw against Montrose.