The home side went ahead after 21 minutes when Paul McManus showed why he is near the top of the second division goal-scoring charts with a clinical volley which flew in via the right-hand post from 12 yards.
The same player almost got a second after 40 minutes, but after racing clear his shot lacked the power to beat David McGurn.
Scott McLaughlin thumped the post from 20 yards as Morton looked for a way back into the match before half-time.
Raith should have wrapped up the points at the start of the second half as they created numerous chances. McManus glanced a header just over, while Brian Fairbairn could only head straight at McGurn from just six yards.
McGurn then made two top saves to deny McManus and Scott Crabbe as Rovers piled on the pressure.
Despite the one-way traffic, the killer second goal never came and Rovers were punished in the last minute as the visitors rescued an unlikely point.
Andy McLaren hit a hopeful shot from 18 yards that took a wicked deflection off Todd Lumsden and sailed over the head of goal-keeper Alistair Brown into the roof of the net.
Rovers had one final chance to get the three points they deserved, but McGurn made another outstanding save in injury time to deny Fairbairn after the winger had latched on to Laurie Ellis's long throw.