The Ukrainians reached the quarter-finals for the first time with a 3-0 win in their home second leg on Tuesday but Borriello's 28th-minute penalty miss at 1-0 was a turning point given Roma had started the game well.
With usual penalty taker and captain Francesco Totti surprisingly kept on the bench all game, David Pizarro was the assigned player having scored a spot-kick in Friday's 2-1 Serie A win at Lecce but Borriello decided to shoot himself.
"I assume the responsibility for having missed the penalty, I wanted to take it, it could have changed the game," Borriello told reporters before having a little dig at new coach Vincenzo Montella, who replaced Claudio Ranieri following the first leg.
"Montella took the duties away from me without talking to me. When Totti wasn't there in the past, I've taken them."
Montella was more annoyed with defender Philippe Mexes for being sent off for two yellow cards after 41 minutes than he was with Borriello.
"I can say that we go out with our heads held high. I'm annoyed for other reasons, too much tension, too many pointless bookings. At this level you can't present them with an extra man for 50 minutes," 36-year-old rookie coach Montella said.
"At Lecce, Pizarro was the penalty taker and it was him tonight too. Borriello asked him if he could take it and he let him take it. Penalties can be missed.
"Next time I'll take it," joked the interim boss, nicknamed the 'little aeroplane' for his celebrations while Roma forward.
The fact the coach was able to joke shows the mood in the camp is not totally desolate even if fans will wonder why their talisman and Montella's former strike partner Totti never saw any action if he was fit enough for the bench.
Roma, irrespective of the 3-2 first leg loss in Rome, were never going to win the competition and their exit allows Montella to concentrate on trying to propel the side further up Serie A and qualify for next season's tournament.
The cash-strapped club, who could be bought by American consortium Di Benedetto soon to receive a financial boost, are sixth in Serie A after two wins and a draw under Montella and are five points behind fourth-placed Lazio ahead of Sunday's city derby.
