West Ham forward Andre Ayew scored from the spot in the 32nd minute following a foul on Asamoah Gyan to hand the Black Stars a 1-0 triumph at Stade de Port-Gentil.
The 2015 Afcon finalists were slammed back home for a poor showing against the Cranes in second half.
"I'm happy with the three points," Grant said.
"We could have played better but winning is the most important in tournaments like this.
"We had to take on the pressure in the second half but at the end of the day, I am happy.
"Uganda is a very tough team and playing against them is never easy.
We got a few chances but we won."
Ghana had failed to beat Uganda in their last three meetings in all competitions prior to Tuesday's encounter.
The Black Stars face Mali in their second group game on Saturday.
"The second game will be different but there are no easy games in this tournament," Grant added.
"It is only the first game and the most important is to win. Andre and Jordan Ayew had a good team.
"In the second half, we were a bit fatigued and the players wanted to keep the three points, and we played in a clever way to win the game."
Ghana now top Group D, two points above Mali and Egypt, who shared the spoils following a scoreless draw in the second fixture.
Uganda lie bottom without a point after matchday one.