Palace, whose players have not yet been paid their December wages, put their off-field troubles behind them to take a 19th-minute lead through Neil Danns.
Although the Owls drew level thanks to Palace defender Clint Hill's own goal just before half-time, Andrew booked the Londoners a place in the fourth round with a well-taken 68th-minute winner.
Wednesday chairman Lee Strafford, who sacked manager Brian Laws in early December, had urged fans to get behind their side in the build-up to the third-round tie, but there was little response from the blue and white half of the city after a run of 11 matches without a win prior to kick-off.
That has now been stretched to 12 in all competitions, with the Owls without a victory since a 2-0 success against Coventry on October 17.
Palace dominated the first half-hour after taking the lead through Danns' opportunist strike.
The former Birmingham midfielder pounced on a defensive lapse on the edge of the six-yard box and swept the ball home, right-footed, into the bottom corner in the 19th minute.
Danns volleyed wide from 20 yards soon after grabbing his fourth goal of the season, while midfield team-mate Darren Ambrose had gone close when trying his luck from long range in the fifth minute.
Wednesday had mustered just one chance on goal up until Palace's breakthrough, with striker Francis Jeffers, back in the starting line-up for the first time since being sent off in a Carling Cup defeat at Port Vale at the end of August, seeing his angled drive blocked by Palace defender Jose Fonte.
But the Owls were back on level terms a minute before the interval. Jeffers and Leon Clarke had successive shots blocked and following the latter, Palace defender Hill inadvertently turned the ball beyond his own goalkeeper Speroni from six yards.
Wednesday wrestled back an equal share of possession after the break, but still found chances hard to come by, while Palace were dealt a blow when Scannell was forced out of the action through injury and replaced by Andrew in the 58th minute.
But Palace were untroubled by Scannell's loss and it was Andrew who regained his side the lead following more hesitancy at the back from Wednesday.
Owls defender Frank Simek failed to clear a long ball forward on the edge of his penalty area and Andrew nipped in to steal the ball and fire a low left-footed shot into the bottom corner.
Jeffers, Tudgay, Jermaine Johnson and substitute Etienne Esajas all had shots blocked or saved in the closing stages, but Wednesday caretaker manager Sean McAuley was left chasing his first win after three matches in charge.