The Cottagers were given a stern test by their depleted Coca-Cola Championship opponents and needed a second goal from striker Andy Johnson three minutes from the end of normal time to secure their first away win of the season.
Hodgson, who made just two changes to the side that drew with Chelsea, has now guided Fulham to a 10-match unbeaten run and is hoping to combine a climb up the Barclays Premier League table to go with a good FA Cup run.
"I'm delighted we've pulled that off," he said. "It's our first away win of the campaign and it was important to get it.
"It was a hard-fought and very thrilling Cup-tie and all credit to Sheffield Wednesday for the way they played today and forced us to give of our best.
"We had a good spell of play in the second half but I was wondering if the second goal would actually come because the goalkeeper made a couple of good saves and we hit the post a couple of times but luckily it did come.
"In terms of a cup run, I would be lying if I said it was as important as survival in the Premiership but it would be nice if we could do both."