The defeat at Amex Stadium ended the team's Cup hopes in just the second round stage, extending a disappointing run which has seen the Black Cats go no further than the fourth round since 2000-01.
The result comes after Sunderland scored just once in their first two Premier League matches, with the lack of goals a factor which is concerning Bruce.
Speaking to BBC after the game, Bruce stated: "The difference between a Premier League and a Championship team should be the ability to take chances, so it's a big concern.
"It's not acceptable to keep creating glorious chances and not taking them.
"We've created enough chances, with the quality that we have got, but at the moment it's our Achilles heel, and we've been punished.
"We've come to take it seriously because we knew what it meant.
"Unfortunately we haven't been good enough on the night to take our chances and win us the match."
Sebastian Larsson's stunning equaliser against Liverpool on the opening day remains Sunderland's only goal this season, with last weekend's derby against Newcastle ending in 1-0 defeat.
This week, the Black Cats return to Premier League action on Saturday with a visit to Swansea City.