Chiefs were hurting prior to the game following their first League loss of the season on Saturday, a 3-1 reverse to Soweto rivals Orlando Pirates at FNB Stadium.
However, they responded by beating Arrows at the same venue via a brace from Man of the Match George Lebese and goal from Bernard Parker.
Komphela, who replaced Stuart Baxter in the off-season at Naturena, was quick to point out his appreciation for the supporters at Soccer City.
"It was a very simple match, yet very complex," he said.
"Number one, I want to compliment these beautiful people that came here tonight. There's an emotional banner they had there and I had to share the message with the players.
"The banner was reading something to the effect of: 'A supporter who comes to a match on the back of loss in a derby is a real supporter.'
"That is a massive statement. When you read that and it sinks... it makes you emotional.
"We thank these people. It aint easy, no-one said it would be easy anyway but we will soldier on.
"Regarding the match, we could have scored more goals. First half we could have gone in the dressing room with more goals. But we will appreciate what the All Mighty has presented tonight, even though we conceded tonight. I just hate condeding goals, it happened yet again. We just have to keep on working hard."
Meanwhile, Komphela says his team have an opportunity to show their character when they revive hostilities with Bucs in the Telkom Knockout semi-finals this weekend.
“It's another opportunity in seven days to redeem ourselves, yet another opportunity to rectify our mistakes. It is not about the number of times you go down, but coming up one more time is the difference. Characters are built and characters will be revealed.”