The Blades head into their last three games five points adrift of second-placed Bournemouth and four behind third-placed Brentford with a game in hand on both.
"We've got three games remaining and I think we'll be sitting in the dressing room before the Preston game knowing a win will give us a chance of automatic promotion," said Morgan, who replaced Danny Wilson as boss last week.
"I might be proved wrong, but I'm optimistic we can take the race to the final day of the season.
"The top two remains in our sights and the results in the next week or so are going to have plenty of twists and turns, I'm sure of that."
A crowd of 23,431 watched the Blades draw with Brentford at Bramall Lane on Tuesday night and Morgan said the fans have a big role to play during the run-in.
"I knew before Saturday we would get the backing of the supporters," Morgan added. "They turned out in numbers against Swindon and to get over 23,000 against Brentford is absolutely incredible really.
"We need them to stick with us. They believe we can win promotion and our belief is as strong as ever."