Looking to use the Scottish Cup to kick-start a faltering season, Inverness started brightly and could have been three up inside the opening 10 minutes.
Roy McBain was first to try his luck from long range with an effort that Jon Tuffey turned over the crossbar, before Andrew Barrowman was inches away from getting his head to a cross from Morais.
Barrowman, however, passed up the best opportunity of the match on 10 minutes, latching on to a through ball from McBain but, with Tuffey racing out of his goal, he slid an effort narrowly wide from an angle.
First division promotion hopefuls Partick had posed very little threat up to this point, although Liam Buchanan did test Caley Thistle goalkeeper Michael Fraser - making his first start since the 5-0 loss to Rangers at Ibrox last November - with a shot from the edge of the are after 11 minutes.
This helped to steady Thistle nerves and midfielder Gary Harkins went close seven minutes later with his 25-yard strike skidding just past the post.
These two efforts were about all the Jags could muster and Inverness made their superiority pay on 27 minutes.
Morais showed just why Inverness manager Brewster had brought him to the Highland capital by taking a couple of neat touches before finding the corner of Tuffey's net from 22 yards.
That was about it in terms of action for the first half, but the second half was barely a minute old before it burst to life.
Don Cowie's pass picked out the run of Iain Vigurs and from just inside the box he unleashed a first-time shot that arrowed past Tuffey and into the top corner.
Caley Thistle, who went into the game on the back of a six-match losing run in the SPL, looked now to be heading to a rare home win, but Partick came close to pulling a goal back on 49 minutes.
Kevin McKinlay set up half-time sub Scott Chaplain has shot appeared to be creeping in at the near post until Fraser stuck out a glove to push it wide.
But Thistle's comeback hopes were ended on 56 minutes as Morais ran on to a through ball before forcing it pas the onrushing Tuffey for his second of the game.
The game then petered as Caley Thistle eased themselves to a much-needed victory.