Two-time defending champion New Zealand saw an unbeaten run at Rugby World Cups that stretched back to 2007 ended in convincing fashion by England.
Eddie Jones's side ran out a 19-7 winner in its semi-final to consign the All Blacks to the bronze-medal match.
Kiwi scrum-half Smith was adamant the reverse was not for a lack of heart as he compared the New Zealand dressing room to a funeral.
"I'm truly gutted and highly embarrassed," Smith said. "You've got family, friends texting you, but you know they're pretty gutted.
"If New Zealand [the country] thinks that we're not gutted, you've just got to go see that changing room. It was like a funeral.
"We are putting on brave faces so it's going to be a long summer. It's over and we trained our guts out. We trained hard for this and prepared well.
"But, in the end, sport happens and we got beat."
New Zealand winger Sevu Reece paid tribute to England, which can look forward to next Saturday's final.
"They were the better team in the end," Reece said. "They came out really strong at the start and they managed to hold onto that for the whole 80 [minutes].
"We expected them to come out with the performance they did but credit to England for a great performance."