The Light Blues, who were knocked out of the League Cup by St Johnstone in midweek, looked to threaten early on, with Gedion Zelalem finding space to shoot on the edge of the box, but his effort was comfortably gathered by keeper Derek Gaston.
Waghorn stepped up and sent his spot kick beyond the reach of Gaston to score his 12th goal of the season.
On 15 minutes, Oduwa again caused problems for the Morton defence as he tried to flick the ball to Zelalem, but the Greenock backline were able to clear the danger.
Waghorn then doubled his side's lead at Cappielow Park when the ex-Wigan Athletic man sent his effort across Gaston into the corner.
Moments later, Morton almost pulled a goal back with a sweetly-struck free kick from Forbes, which beat Wes Foderingham, but crept the wrong side of the post.
On 34 minutes, Rangers all but secured victory as Tavernier netted his eighth goal of the season.