The decision to award England a penalty after 102 minutes of play at Wembley stood up to a VAR check and Harry Kane had the spot-kick saved before he buried the rebound past Kasper Schmeichel.
Earlier, England had fallen behind to a superb long-range free-kick from Mikkel Damsgaard before Sterling forced an equaliser that went in off Simon Kjaer.
Sterling said he felt the decision to award the penalty in extra time was correct, saying: "I went into the box, he stuck his right leg out and it touched my leg so it's a clear penalty.
"As long as it goes in the back of the net, that's all that matters."
Sterling has scored three times on England's route to the final at Wembley, where they will play Italy on Sunday evening.
The Manchester City forward said the experience of bouncing back after conceding their first goal of the tournament would stand England in good stead against Roberto Mancini's Azzurri.
"It was a top performance," said the 26-year-old. "We had to dig in deep. "It was the first time we conceded but we responded well and showed good spirit.
"We knew it would be difficult. We stayed patient and we knew the legs and aggressiveness we have in the team we'd be okay.
"It's another step in the right direction. We have to focus on the weekend now. It's step-by-step. We know what football means to this country. The energy, the atmosphere...it was top.
"Now we have Italy. We will celebrate a little bit then focus on Italy."