Furman put in a man-of-the-match display in the goalless draw with Libya in a 2019 AFCON qualifier on Saturday, although he received some jeers from fans at Moses Mabhida Stadium on his 47th appearance for the national team.
Furman, who offers security in the engine room with his workrate and ability to read the game, almost capped off the showing with a goal as he hoisted a lob onto the roof of the net to nearly expose Libya keeper Mohamed Fathi early on.
After fielding questions at the post-match press conference, Baxter applauded the SuperSport United captain and Keagan Dolly, and questioned why the former sometimes "takes stick" from the media or supporters.
"I think one thing that comes to mind is the fact that I thought Dean was outstanding this evening. I thought Keagan Dolly was excellent as well," said Baxter, who was sitting alongside Furman.
"I know that when we select the team there's always someone that pops his hand up and says, 'What about Dean Furman and why is he playing? What about Keagan Dolly, why is he playing?'
"When you make that sort of comment I mean people must feel pretty silly when he puts in the performance he did tonight. Dean is an asset to South Africa and I think anybody that doesn't see that doesn't know football.
"It is not criticising any individual sitting in here but that is just me and that's just ... we all need to take responsibility for the development of this game and constant negativity to certain players or the whole of South African football doesn't encourage anybody.
"So as Dean says, we know what's gone down tonight. We know we should have won. We are disappointed and we will analyse and try and come back stronger.
"But I have to say that players like Dean, he puts his heart and soul into every game and I think we all need to ask ourselves a question, are we fair enough with the criticism or maybe not?
"I feel Dean does take, for me, a bit of unwarranted stick and I thought he was magnificent tonight."
As the press conference was ending, the matter of Baxter hiring his son Lee as a goalkeeper coach for the game was brought up again. Baxter recruited his son after keeper coach Andre Arendse pulled out of the camp two days before the squad headed to Durban after his wife faced complications from an operation.
Asked about his reported comments that people who don't know football are questioning the decision, Baxter said he was misquoted.
"I didn't say that people don't know football. What I said, if you listened to me, was I don't understand any criticism of me bringing in Lee in the camp. He's the most qualified goalkeeper coach in this country.
"He's more qualified than many of the PSL goalkeeper coaches so I don't say you don't know football. I'm just saying I'll be surprised and very disappointed if we don't go above that. That's not me.
"It's the same that they don't know what they're talking about, I didn't say that. I said, I'll help them and people to understand that the negativity they create when they lambaste a warrior like Dean Furman, they have to understand what they do. They have a part, as you have a part now, saying that I'm telling the nation they don't know football. You are misleading."
When journalist Thabiso Mosia suggested that he's got the audio of Baxter saying people in this country don't know football, he responded: "I hope you got it on record my friend because I didn't say it."