On a night when Wednesday would have been relegated to the third tier for the first time since 2000 if they had lost to Rovers and Derby have beaten Preston, the Owls held their nerve and triumphed thanks to a goal eight minutes before the break from Josh Windass.
Elsewhere, North End did them a big favour by beating Derby 3-0. With three games to go, the Rams hold a four-point advantage over second-from-bottom Wednesday, who visit Pride Park on the last day of the season.
Preston's victory was also good news for the side occupying the final relegation position in Rotherham United, who welcome Neil Warnock's Middlesbrough on Wednesday evening.
The Millers are level on points with the Owls - with a better goal difference - and also four points behind Derby. But Paul Warne's side have three games in hand on the Owls and Rams, starting with the game with Boro.
Smith said: "It just keeps his belief and we have always been positive. There are three games to go and we hope to win the three games and we will do all we can to do that to get nine points and see where we are.
"We have got to win these three games, it is as simple as that and we will be doing everything we can and not worry about what other people are doing and just worry about ourselves. Our results will take care of themselves and if we start worrying about other things, it will detract from what we are trying to do and we will concentrate on ourselves."
On a solid and professional performance from the Owls, who were defensively strong all evening, Smith continued: "I was delighted and we were really disciplined throughout the game and scored a good goal. We have a lot of leaders out on the pitch and you want them to be organised. It cannot always come from the sidelines and there was a lot of experience out there and it was good to see them demanding of each other.
The result will have also lifted the spirits of head coach Darren Moore recuperating at home and out of hospital following treatment for pneumonia triggered by his contraction of Covid-19.
Smith added: "He is a lot better today. I spoke to him before the game and he watched the game and he was in contact tonight which was pleasing and nice for us.
"It is a little bit too early to say (whether Moore will return to the touchline) this season and we are still out of the woods. But he is a lot better and improving."