According to the Football411 website, Amakhosi entered Saturday's tie having cruised to a 3-0 win away to Phunya Sele Sele in the PSL a fortnight ago, and Lincoln Zvasiya's deflected header this weekend gave the Soweto giants their place in the last-four.
Speaking after the match, Baxter told his club's official website: "I am proud of the players. It is not easy to come here (to Bloemfontein) and win two games in a space of two weeks.
"Credit to (Celtic) coach Clinton Larsen as they pushed us until last minute forcing a superb save from Itumeleng Khune. They were very organised and kept on going."
Questions were asked about the state of the pitch at the Free State Stadium after the tie, with Baxter admitting the conditions were well below par for a top South African stadium.
He added: "I sympathise with Larsen who has to play on this pitch week in week out.
"These are the two teams that enjoy playing football and the condition of the pitch did not really help."
And while Baxter was more than happy to celebrate the quarter-final success, the Englishman is looking to quickly shift the squad's focus to the PSL, where they travel to the Dobsonville Stadium to face rivals Moroka Swallows on Saturday.
"We just have to focus on our game and of course this weekend we have a big game, a Soweto Derby against Moroka Swallows in the league," added Baxter.