On-loan striker David Witteveen scored a hat-trick to earn Morton a point as they shared six goals with Queens.
Rocco Quinn was on the end of a Derek Holmes pass to fire Queens ahead then Witteveen prodded a leveller despite David Hutton's claims for a foul.
Witteveen hammered home a Ryan McGuffie cross to put the Ton ahead before Quinn added his second in off the post.
David Weatherston sprung the offside trap to give Queens the lead again only for Witteveen to level in injury time.
The point for the Greenock side could prove to be vital on the final day of the season, with bottom club Airdrie beating Partick Thistle and putting together a run of form that may yet see them catch relegation contenders Morton, Ayr United or Raith, though Rovers have a few games in hand.
After Quinn had put the visitors ahead from inside the box, there was a moment of controversy when Queens goalie Hutton claimed that Witteveen, on loan from Hearts, had knocked the ball from his grasp to prod home the equaliser.
For the Austrian's second, there was no debate as he met McGuffie's cross from the right wing and shot on the turn to send a low shot into the net.
A Holmes lay-off set up Quinn for his second as he went racing through on goal and managed to steer the ball past Colin Stewart off the base of the post.
Sub Weatherston, on the pitch for only 30 seconds, used his pace to beat the offside trap and fire inside Stewart's post to give Queens the lead again with 25 minutes to play.
In injury time, Jamie Adams missed a great chance to make it four for Queens but blasted over, then with seconds remaining the hosts made their way up the park and Witteveen scored his hat-trick by slotting under substitute goalie Scott Fox to round off the scoring.
Earlier in the match, Fox, who had come on for David Hutton after the Queens number one hurt his back in the warm-up, pulled off a fantastic one-handed save to deny Peter Weatherson.
And Colin Stewart had a great one-handed save to deny a close-range header from Adams.
This was a case of three points being thrown away. Queens made enough chances to win comfortably and Jamie Adam's miss was costly.
Queens Mad Man of the Match - Rocco Quinn - Took his goals well and was unlucky not to score a hatrick.