Bersant Celina’s 25-yard strike midway through the second half was enough to seal a first victory since mid-December for the Blues, who remain 12th in the Championship table but are only four points off the play-off spots.
Leeds saw Eunan O’Kane sent off in the 37th minute following an altercation with Town left-back Jonas Knudsen.
The Whites have now taken one point from their last three games and stay just outside the play-off places in seventh.
McCarthy said: “I thought it was a really tough game against a team that’s in good form with good players.
“I haven’t seen the sending off, someone said it was a headbutt. If that’s the case then he deserves to be sent off. That doesn’t make it easy, we’ve scrapped away.
“We needed the result over and above anything because we’ve had three defeats on the bounce and we needed to get our fans back on side here.
“I think their fans have helped more than anything else because they were brilliant and they created an atmosphere that was brilliant to play in and to manage in.
“They’re a good side but we’ve had to scrap it out and we’ve played well, (I’m) really pleased.”
Speaking about Celina, McCarthy said: “Sometimes he’s causing us as many problems as he’s causing them, losing it, but that’s what you get with him.
“He works his socks off. He’s a proper little character on the pitch. That one little bit of magic in the second half wins us the game.”
Leeds boss Thomas Christiansen felt his side deserved more from the match.
He said: “I’m…