Substitute Nikola Zigic pulled one back for Birmingham after the break.
But man-of-the-match Jermaine Johnson raced away for Wednesday's third, before Marlon King scored a penalty in stoppage time.
Craig BurleyBBC Radio 5 live
"Sheffield Wednesday are good movers and have some powerful players. They were very pacy down the left and Jermaine Johnson was the best player on the pitch.
"Birmingham were not quick enough and sharp enough in getting to the second balls in the first half. They didn't do the ugly side of the game."
Wednesday, promoted from League One last season, have won 12 matches and drawn four since Jones took over in March.
They deserved victory over a Birmingham side who were sluggish in the first half, but were helped on their way by a Jack Butland mistake.
Birmingham's Butland, who became England's youngest-ever keeper when he made his debut at the age of 19 against Italy last week, dropped a Jose Semedo cross under no pressure, and Reda Johnson headed into an empty net for his second goal in two games.
Then Rodri latched on to a well-timed through ball, fooled backtracking Blues defender Steven Caldwell with a clever step-over and fired the ball left-footed beneath a helpless Butland into the bottom corner.
Birmingham manager Lee Clark brought on Serbian striker Zigic at half-time, and saw an improved performance from his side.
Wednesday keeper Chris Kirkland, who had earlier tipped away a Caldwell effort, did well to keep out King's half-volley.
Jermaine Johnson raced clear within two minutes remaining to fire past Butland before King converted a penalty with the last kick of the match, awarded after Chris Burke had been dragged down inside the area.
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