He struck with a 25-yard shot eight minutes into the second half after Dunfermline had failed to clear a corner.
It meant Caley finished the season with five successive wins since the league split without losing a goal.
It was hardly inspired stuff. Bartosz Tarachulski failed to connect with a header early on after good work from Noel Hunt, but neither goalkeeper was asked to over-exert himself.
Bryn Halliwell looked uncomfortably at a Morgan cross which dipped just over his crossbar as Caley Thistle looked the more likely to make the breakthrough.
Barry Wilson had the first shot on target midway through the first half but it was a tame effort from the edge of the area and Halliwell gathered easily.
Dunfermline had a worthwhile effort themselves after 27 minutes when Greg Shields nudged a header just a foot over the top from a Scott Muirhead cross.
Caley responded and Ross Tokely should have done better than send a header high over the top after outjumping the Dunfermline defence to meet Morgan's corner.
Seven minutes from the break, Halliwell was caught out by a flashing header from Graham Bayne but, fortunately for the goalkeeper, it drifted over his crossbar.
Rory McAllister brought the save of the match from Halliwell with a rising shot three minutes into the second half.
But Caley Thistle struck the decisive goal a few minutes later. Nicky Phinn headed out a Wilson corner, but Morgan pounced to rifle in a low shot from 25 yards.
Dunfermline tried to respond and Mark Brown had to be on his toes to tip over a header from Noel Hunt.
But it was the Inverness side who looked more likely to add to their tally on the break.
Morgan shot straight at Halliwell and Bayne should have done better than shoot past from 16 yards with the Dunfermline defence pulled apart.