Willian scored the decisive second as Chelsea advanced to the quarter-finals but they were pushed to the final whistle by the rejuvenated visitors.
Chelsea had contained Everton with ease in the first half and led through Antonio Rudiger’s looping header.
But the Toffees were sharper after the break as Kevin Mirallas had a shot well saved and Ademola Lookman hit the bar.
Willian drilled a superb low finish home in stoppage time before Dominic Calvert-Lewin netted a consolation.
“I’m a bad loser but I have to say I’m proud of the performance, they were terrific,” Unsworth, who has replaced the sacked Ronald Koeman for now, told BBC Radio 5 live.
“It was a committed performance. Hopefully we’ll show that in every game I am in charge.”
