New signing Rade Prica marked his debut with a vital goal as Roy Keane's side lifted themselves out of the relegation zone.
Daryl Murphy had fired the Black Cats into a 15th-minute lead as the home side deservedly took three priceless points.
"Any team that comes to us at home, we definitely do well. It's about getting every point we can from now until the end of the season and hope for the best."
Goalscorer Prica said: "Of course, my first game and scoring, it can't be better.
"I saw it long before it was headed to the goalkeeper and it was a nice goal."
Asked whether his disallowed effort should have counted as his second goal, Prica said: "I think so. But the referee didn't think so. Maybe there was a little hand!"
Sunderland boss Roy Keane said: "What pleased me most was the fact we won the game without playing particularly well. I'm sick of games when we've done really well and lost."
Keane was especially impressed by Jones once again, saying: "He was massive again, with the important goals he's scored for us, defensively he's done very well for us, he's a real handful.
"He's been a major plus since he's come to the club, been delighted with him.
"It was an important result tonight. I've been saying all week we can't look too far ahead."
Birmingham boss Alex McLeish was understandably unhappy with his side's performance.
"With the second ball, Murphy has not been picked up when there were plenty of bodies in there and that's disappointing," he told Sky Sports 2.
"In the second half, losing the second goal because of a bit of indiscipline, trying to get the goal back we left ourselves very exposed. We were lucky we didn't lose by more.
"Sometimes you've just got to go for the second ball but we didn't cover."