Matsatsantsa sneaked into the top eight despite a 2-2 draw against Orlando Pirates at the Orlando Stadium on Saturday afternoon.
The Tshwane side were tasked with having to beat the Buccaneers in their final league match in order to have hopes of qualifying for next season's MTN 8.
However, a draw was enough to leap frog Golden Arrows, who fell to a 2-1 defeat against relegation battlers University of Pretoria.
Baxter was overjoyed at his side's achievement after their dismal start to the season, in which they have now managed a top eight finish as well as a Nedbank Cup final.
"We were dependant, obviously, on Tuks' result so we didn't come all guns blazing from the beginning," Baxter explained to SuperSport after the match.
"We tried to play a bit smart. We thought we could more from Jeremy when he went on. We conceded from a set play – a good goal from Pirates – and we got ourselves back into it.
"We then started playing better football and thought we were through at 1-1, and then they scored and we really had to throw the kitchen sink at them in the last five minutes."
The former Kaizer Chiefs gaffer paid tribute to his players for showing character to fight back and showing their desire to finish in the top half of the table.
"Massive credit to the players because they showed, again, huge character and wanted to be in the top eight."