The Black Cats bossed the ball in the first half without making much headway in terms of chances until a mistake by Sam Hutchinson in first-half stoppage-time allowed Ross Stewart to open the scoring.
But when the final whistle blew, the tie was still far from lost, despite the hosts' dominance.
"I don't think we started too conservatively," said manager Moore. "Right from the word go we knew it was going to be that way. They came at us and that's play-off football for you.
"Honestly, it was exactly how I thought it would go.
"We knew we had to keep the crowd quiet and just before half-time your speech is a certain and the next minute it's about not letting the disappointment seep into the second half.
"You go in at 0-0 and the onus is on them but at half-time we had 15 minutes to settle them down and regroup.
"For the first 20 minutes of the second half they came at us again but after 60-odd minutes I felt the pendulum swing."
The Owls rode their luck at the start of the second period, Alex Pritchard's shot against the crossbar the best of a flurry of opportunities. Even when the visitors took back a measure of control, though, they were unable to work Lee Burge and Stewart left them off the hook with a missed chance after 89 minutes following a mistake by Bailey Peacock-Farrell.
With only one goal in the tie and a sold-out Hillsborough, where the Owls' form has been excellent this season, awaiting on Monday there is still everything to play for.
Sunderland struggled in possession with Barry Bannan playing 90 minutes after an injury scare in the build-up but not having his normal influence.
Moore, as usual, was very level when assessing the first leg.
"We knew it was going to be a tough game," he said. "I've played loads of times here as a player and come here as a manager and when the crowd gets up like that it's a difficult place.
"The disappointing thing for me about the game was we didn't use the ball as well as we could but give Sunderland credit, they put a lot of energy into the game, worked, pressed us and used the ball well.
"It's not easy here with a full house under the lights.
"The positives were that the boys were dogged and put their bodies on the line.
"The goal was avoidable and we accept that but after about 60, 65 minutes I just felt the energy on the pitch change and we finished the game a lot stronger without looking really that threatening. We could have used it better.
"The game lived up to my expectations, we knew we had to stay dogged.
"I said to them at half-time we don't concede again, make sure you do your jobs and stay in the game because we'll get our moments. That's how it panned out."
On his response to Hutchinson's mistake, dallying on the ball to let Stewart in, Moore said: "He was one of those I said, 'Come on, it's gone now,' to. We were all like that because we score as a team, we concede as a team.
"We knew it was avoidable but in the second half he put in a wonderful, spirited performance and he did his job in the second half. We grew in confidence and he was trying to get himself forward to get a goal."
Opposite number Alex Neil was delighted with Sunderland's performance.
“I thought we were excellent," he said. "To handle such a high-pressure environment – the atmosphere was electric, so credit to the fans, I thought they were an absolute credit to the club tonight - it was brilliant.
“But the players have got to then take that take that expectation, take that atmosphere, soak it up and then try to manufacture a really good performance which I think they did.
“The only disappointment from our point of view was probably the fact tat we didn't add to the scoreline. Particularly from half-time to about 75 minutes, we probably had two or three glorious chances to double the scoreline, and it certainly wouldn't have been any more than we deserved at that point."