Yesterday's 169-run success in Cardiff saw Cook's men go 1-0 up in the five-match series and was an ideal way for England to mark their first match under new Australian coach Trevor Bayliss.
Australia, set what would have been a new Ashes fourth-innings record winning total of 412, were dismissed for 242 just over an hour after tea on the fourth day.
England paceman Stuart Broad produced the latest of several decisive Ashes hauls in taking three for 39 on Saturday after man-of-the-match Joe Root had got the hosts going with 134 in a first-innings total of 430.
"It's really important for this group of players to show that we can compete with Australia," Cook said.
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Although it is 14 years since Australia won an Ashes series in Britain, there had been speculation they would have a mental edge over England following a 5-0 whitewashing of Cook's then side during the previous Test contest between the two sides 'Down Under' in 2013/14.
However, England's team in Cardiff featured four Ashes debutants in opener Adam Lyth, wicket-keeper Jos Buttler, off-spinner Moeen Ali and fast bowler Mark Wood.