The tie was as good as over after only 10 minutes courtesy of a curling effort from Jacques Maghoma and Gary Madine's close-range finish.
Atdhe Nuhiu made sure of Wednesday's progress with a header from six yards midway through the second period.
The Owls have now won two out of two to kick-off the new season in style, in contrast to their dismal start to the 2013/14 campaign.
Gray feels there is even more to come from his men as they look to improve on last term's 16th-place finish in the Sky Bet Championship.
"I said to the players beforehand, 'Start brightly and make sure you come out of the traps'," he said.
"I sensed it in the warm-up - they were excellent, zipped the ball about and took it into the game.
"These games can be a potential banana skin but I thought the timing of the (first) goal was good for us.
"(But) I'm sat here disappointed again that we only won 3-0 when we've created umpteen chances, ones we've got to be converting."
Wednesday face a sterner challenge when they return to league action on Saturday, with last season's play-off finalists Derby the visitors to Hillsborough.
New signing Stevie May is available after missing the cup date with the Magpies through suspension, leaving Gray with a selection dilemma.
"It wasn't until late last night that we found out Stevie May was suspended. He was going to start," he added.
"He'd have looked there tonight and thought there was a great opportunity for him.
"You've got to have that competition and both strikers have scored, and Jacques Maghoma has, so it will be a difficult decision to make on Saturday against Derby."