Kristijan Kahlina recorded six saves in the shutout as Charlotte avenged a 2-1 loss last month in Atlanta.
Charlotte (3-4-0, 9 points) got a boost from the early goal and sustained that momentum for long stretches.
Atlanta (3-2-1, 10 points) seemed unable to overcome lineup changes necessitated by a recent rash of injuries.
Alcivar's outstanding "Olimpico" goal came as he launched a right-footed shot on the set piece, with the ball finding the net in the top right corner. The midfielder put the ball just over the outstretched hand of Atlanta goalkeeper Brad Guzan.
Guzan finished with three saves.
After being shut out in its previous game, Charlotte made the early strike hold up.
Atlanta threatened at times during the second half, particular during a stretch midway through the period.
Still, Charlotte had the better of the play for much of the second half.
There were four yellow cards in the final 20 minutes in a game that became physical. There were 26 fouls called in the game, with 17 of them on Charlotte.
This was the only home match in five April assignments for Charlotte, which improved to 3-1-0 at home in its first season in MLS.