Knapp downed Riske 6-3, 7-5 in a 1-hour, 37-minute match, giving the Italians an insurmountable 3-0 lead in the best-of-five opening-round series at Cleveland Public Arena.
Riske took the court for the United States instead of the scheduled Madison Keys, who had been trounced by Italian Camila Giorgi in Saturday's second match.
The victory put Italy into the Fed Cup semifinals for a sixth straight year. They have won the title four times since 2006.
Their opponent in April's semifinals will be the winner of the Spain-Czech Republic first-round match. Due to rain delays, that matchup is set to be concluded Monday in Seville, Spain.