Eduardo Vargas scored a double to fire Chile to a 2-1 win at Santiago's National Stadium over a gallant Peru, who had Carlos Zambrano dismissed early in the first half.
Chile will play the winners of Tuesday's semi-final between Argentina and Paraguay for a shot at their first title in the tournament's 99-year history.
"It was our obligation to get Chile to the final, we knew it from the first moment," Sampaoli told reporters.
"It's an incredible dream and even more so as hosts."
Sampaoli said nerves had made the win more difficult than needed, even with Chile one man up for most of the match.
"Confusion complicated us in a game that we should have secured much earlier," he added.
"Of all our Copa games this was the most difficult because we didn't know how to define."
Peru coach Ricardo Gareca praised the hosts, saying Chile were "a firm candidate (to win the Copa), it was from the beginning and continues to be."