Smith was stretchered off with five minutes to play of normal time after Reading playmaker Yann Kermorgant caught him on the ankle with a slide challenge from behind.
The 25-year-old returned to the bench on crutches in extra-time and joined in the celebrations once Chris Schindler’s match-winning penalty hit the back of the net.
The defender, who appeared with his foot in an orthopaedic boot, will have an x-ray on the injury this morning.
“It’s a bit sore so we will have to see,” Smith said.
“It was really tough to have to come off from that game because the atmosphere was incredible, and for what was at stake. But I’m just delighted we got over the line in the end.
“It’s been a phenomenal season for the club. Everybody has questioned us this season. ‘Can they do it?’ And we’ve done it. We’ve stuck together as a team.”
With club captain Mark Hudson on the bench, Smith was given the honour of leading out the Terriers at Wembley. “It was a proud day and a memory that will stay with me forever,” said Smith.
“You always dream, don’t you? But it doesn’t always happen. We thoroughly deserve to get there.”
The defender has been at Huddersfield since his first shot at the Premier League ended in the youth ranks of Manchester City in 2010.
Since arriving at the John Smith’s Stadium, he has made 148 appearances, but he has flourished under David Wagner.
Smith added: “David’s been a key element of what we have done. He made a lot of changes when he came into the football club. The players have bought into it and it’s been a monumental effort from everyone.
“To get to Wembley, and to win it, it’s an amazing feeling.
“It’s a team ethic that I’ve not been involved in before. The team is full of energy. I can’t speak highly enough of the team or the club as a whole. It’s been an unbelievable effort from everyone.
“We are going to go into it with the same attitude as we did this season: no limits. It served us well this year and it’ll serve us well again next year.”