comfortable task of scoring 231 in more than two days for victory.
It was Australia's fourth consecutive win over England and comes four months after they lost the last series 3-0 in England.
"The bottom line is we haven't been good enough," Cook said.
"The part of this game that makes it even more frustrating is that we got ourselves into a good place to put some pressure on Australia: 100 ahead and no wickets down in the second innings.
"I suppose that might be where we are as a side. We created three chances in that first half-an-hour and we didn't take them."
English heads dropped when Cook got his right hand to a snick from Rogers on 19 through slips after wicketkeeper Jonny Bairstow failed to respond, and then grassed a straightforward two-handed chance from David Warner (22) off Stuart Broad.
Warner scored only three more runs before he was caught behind off Ben Stokes and Australia's first wicket came down at 64.
Symbolic of England's malaise, Cook did not bring on his specialist spinner Monty Panesar until 90 minutes into the fourth day and he had little effect on stemming the Australian run flow until he finally removed Rogers.
There will be selection questions ahead of the final Ashes Test for an England team in decline with the most pressure on Panesar, Bairstow, whose technique behind the stumps was exposed, and seamer Tim Bresnan.