After the break, Jamie Campbell played in Chris Erskine, who finished smartly from 25 yards.
Patrick Boyle sealed the win for the Jags, slotting home Martin Grehan's pull-back from the right wing.
Partick: Fox, Kinniburgh, Balatoni, Archibald, Boyle, Doolan, Cairney (Lochhead 88), Campbell, Erskine, Stewart (McGrotty 88), Grehan (Fraser 90). Subs Not Used: Scully, McGeough.
Goals: Erskine 60, Boyle 72.
Morton: Stewart, Evans (Young 82), Smyth, McCaffrey, Monti, Jenkins (McKinlay 90), Bachirou, Tidser, O'Brien (Weatherson 70), Lyle, Kean. Subs Not Used: Cuthbert, Holmes.
Att: 2,060
Ref: David Somers (Scotland).
Source: BBC Sport
Source: BBC Sport