Thembinkosi Lorch's late goal silenced the home crowd inside the Cairo International Stadium and it was enough to eliminate the tournament hosts.
Baxter revealed stopping the supply to the Pharaohs' key attackers Mohamed Salah and Mahmoud 'Trezeguet' Hassan was part of their game plan.
"They worked really hard. We had a game plan (on) how we would try to stop the supply to Salah and Trezeguet," Baxter told the media.
"I think we succeeded in [doing] that. It meant that their best players did not influence the game."
Bafana attackers Lorch and Percy Tau were a constant threat throughout the encounter and Baxter was pleased with his side's attacking play on the night.
"Once we did that we grew in confidence and even our attacking play was much better tonight," he added.
The 1996 Afcon champions have silenced their critics with their impressive display against one of the tournament favourites and Baxter stated South Africans should be proud of his side.
"It feels fantastic to beat Egypt on their home ground. It is a wonderful experience for the boys. I think the country (South Africa) should be proud," he concluded.
South Africa will now take on Nigeria in a quarterfinal encounter at the Cairo International Stadium on Wednesday.
The Super Eagles booked their place in the last eight after stunning the defending champions Cameroon 3-2 earlier on Saturday evening.