After Reda Johnson capitalised on a handling error from England goalkeeper Jack Butland to head the opener, the tricky forward - signed on loan from Barcelona B on Monday - rewarded manager David Jones for handing him his first start with a well-taken finish.
The former Spain Under-21 international should have converted Wednesday's first attack of the match, but made up for his miss six minutes before half-time by wrong-footing Steven Caldwell and firing in Wednesday's second.
Blues manager Lee Clark introduced Nikola Zigic, the scorer of his side's equaliser against Charlton on Saturday, at half-time and the tall striker reduced the deficit 15 minutes from time with a tap-in at the back post.
But Jermaine Johnson netted a late third and, although Marlon King converted a last-gasp penalty, the victory preserved Jones' unbeaten competitive record as Owls boss since taking over at the helm from Gary Megson last season.
With just three minutes on the clock, Rodri almost made an immediate impact. Jose Semedo received the ball down the left from Johnson and provided a cut-back from the byline to Rodri, who could only guide his side-footed effort wide of the post.
Blues started to settle and, after a free-kick on the right from the lively Chris Burke found its way into the path of Caldwell unmarked at the back post, the defender stung the palms of Chris Kirkland.
It was Wednesday who deservedly took the lead after 14 minutes when Olympian Butland, under no real pressure, let an outswinging Semedo cross slip through his hands and onto the head of Johnson, who nodded into the empty net for his second goal in as many matches.
Blues midfielder Darren Ambrose - making his 300th career league appearance - found space on the edge of the box, but could only send a shot high over Kirkland's bar.
Wednesday made their dominance count after 39 minutes when Rodri latched onto a well-timed through ball, fooled the backtracking Caldwell with a clever step-over and fired the ball left-footed beneath a helpless Butland into the bottom corner. The visitors came out firing after the restart, with Zigic leading the line, and King had a half-volley tipped over by Kirkland, who denied the striker again after Burke fired a glorious chance wide moments earlier.
Rodri was given a standing ovation when he was withdrawn on the hour mark, but Blues gave themselves a lifeline 15 minutes later when Zigic latched onto a David Murphy cross to poke home at the back post.
However, Johnson raced clear within two minutes remaining to fire past Butland before King's penalty, after Burke was dragged down in the area, proved to be just a late consolation.