The hosting Whites began the brighter of the two sides in what was Brian McDermott's first encounter at the helm.
However, despite their domination, they went into the break a goal behind courtesy of Wednesday's Jermaine Johnson, who struck in the 27th minute following a mix-up between Leeds defender Stephen Warnock and goalkeeper Paddy Kenny.
McDermott's new side managed to make the most of their chances after the restart, though, and they were able to level up proceedings with 63 minutes played. Ross McCormack sent over a cross for Varney, who rose highest to send the ball over the line via the underside of the crossbar.
The same two players linked again six minutes later for what proved to be the winning goal. On this occasion Varney found himself unmarked and he was able to guide a header beyond the reach of Wednesday goalkeeper Chris Kirkland.
In the closing stages Wednesday had a chance to claim a share of the spoils, but veteran substitute Steven Howard sent a volleyed effort against the upright.
The result has moved McDermott's new side eight points above the relegation zone.