After two inconsequential shots from the home attack which barely troubled Alistair Brown in the Rovers' goal, the visitors came closest to opening the scoring on several occasions in the first half.
Chris Silvestro fired a long swerving effort just wide, and the experienced Derek Ferguson saw a similar shot meet the same fate.
Just before the half-hour, Raith had a splendid opportunity to go 1-0 in front when Ryan Harding mistimed a headed clearance to allow Paul McManus a clear run in on goal, but the striker did a 'Glenn McGrath' on the ball and, although Peter Weatherson clattered his attempted clearance against Scott Crabbe, Chris Millar mopped up the danger.
Neil Jablonski sent McManus clear with a deftly-struck pass but David McGurn dived full-length to save the well-struck drive.
Much to Rovers' exasperation, Morton took the lead eight minutes before the break. Andy McLaren took a free-kick on the right, rolling the ball across the box where Weatherson nipped ahead of Laurie Ellis to crash an unstoppable shot off the underside of the bar and behind the helpless Brown.
In the second period, neither side played with much penetrative attacking verve, but Weatherson and Stewart Greacen had to make splendid last-ditch tackles on Eddie Annand and McManus to preserve the home side's lead.
Six minutes from the end, Millar lobbed over the impressive Tod Lumsden for McLaren to repeat the manoeuvre over Brown to clinch the three points.