Iain Vigurs' goal was sandwich by Morais' brace as the SPL basement side finally found some form with a comfortable victory.
Inverness snapped up Morais from Hibernian this week and he impressed throughout, full deserving his perfect start against the Irn-Bru First Division Jags.
Inverness had lost their previous two cup ties against Saturday's opponents and were keen to avoid a hat-trick of defeats.
The Highlanders fielded new signings Pavels Mihadjuks at centre-half and Morais in midfield, with Lionel Djebi-Zadi replacing Richard Hastings at left-back.
New Partick striker Kris Doolan was on the bench after signing from junior club Auchinleck Talbot.
The wind and rain was howling throughout the Caledonian Stadium, conditions sure to test the players.
It was also going to be telling how the new-look, makeshift Inverness defence would cope with the pace of Liam Buchanan and Simon Donnelly.
Morais joined Andy Barrowman in attack for Caley Thistle, a change from the sole forward role often chosen by boss Craig Brewster this season.
Roy McBain had the first chance when he forced a fine save from keeper Jon Tuffey with a rasping drive from distance.
Then Morais created space on the right and his cross was just missed by Barrowman. The former Ross County man drove just wide in the next attack after he was played through by McBain.
It was not all one-way traffic though and Donnelly tested Michael Fraser with a shot along the skiddy ground, but the keeper clutched it.
A great saving challenge from Ross Tokely came just in time to prevent Kevin McKinlay hitting the target as the Jags ace raced on to a Gary Harkins pass after 13 minutes.
Impressive Harkins shot a warning for the visitors when he flashed a 25-yard effort just past Fraser's right post as Thistle showed what they are capable of.
Partick were forced into a substitution after 26 minutes when John Robertson picked up a knock and was replaced by Mark Roberts.
Almost immediately, Caley Thistle grabbed the lead when Morais latched on to a neat pass from midfield and finished to open his account with a drive into the corner of the net.
The Glasgow men reacted well to the setback and got a few balls into the box, but the Highlanders held firm.
Less than a minute into the second half, Caley Thistle marched into a two-goal lead when Vigurs smashed a great shot high into the net after being played into the box by Russell Duncan.
Substitute Scott Chaplain almost brought Partick back into the tie moments later but his shot was diverted past the post by Fraser.
The rain resumed, joined by the fierce gale, as the hosts took control.
Morais put the game beyond reach after 56 minutes when his perseverance paid off as he prodded the ball past Tuffey from 12 yards after a high ball was played into the area.
On the hour, Fraser pulled off a fine save, denying Donnelly, whose dipping drive was netbound.
With 18 minutes to go, Morais' Hibs team-mate Thierry Gathuessi made his Inverness bow, replacing Duncan in midfield.
Before being substituted 10 minutes from time, Morais continued to entertain the fans with his trickery, but no more goals followed.
When they could, Thistle tried to use their width, but the reshaped Inverness rearguard coped well.