Marouane Fellaini's header gave the Toffees a dream start after a monumental performance from the giant Belgian midfielder, and even the introduction of new signing Robin van Persie couldn't prevent United from slipping to their first opening day defeat since 2008.
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New signing Kevin Mirallas was unveiled before kick-off and is likely to partner the Croatian hitman in attack for the remainder of the season, and Jelavic is confident that his side can build on their encouraging start.
"We can aim high, not only because of this victory but because we have a really good team," he told Croatian newspaper Vecernji.
"We are like one big family and we also signed some great players this summer. You could all see at the United game that we are really strong this season.
"Manchester United are strong but we killed them. David de Gea saved them. We could have really destroyed them. We can be really optimistic."