Cook returned on third morning, 104 not out with England on 192-2 in their first innings and he opening batsmen batted throughout the whole third day to notch up his fifth test double century.
Cook was supported well by Joe Root, who made 61 and Stuart Broad, who made 59 as the tourists ended the day on 491-9 with a led of 164 with two days remaining.
Root failed to convert yet another half-century score
The tourists started the third morning on 192-2, still 135 runs behind Australia's first innings score of 327.
Root, who started the day 49 not out, didn't take long to bring up his half century as alongside Cook the pair looked to kick on.
The partnership though was finally broken on 138 when Pat Cummins removed Root for 61 to tie the Aussies a crucial breakthrough.
Malan failed to review an LBW decision that would have been reversed
The wicket of Root brought Dawid Malan to the crease to join Cook and just as the pair started to get going, Malan was given out LBW to Josh Hazlewood for 14 even though he had inside edged it but decided against reviewing the umpire decision.
Jonny Bairstow came to the crease at the fall of the Malan wicket and alongside Cook the pair saw England the lunch without losing anymore wickets.
Therefore at lunch, the tourists ended the session on 264-4 with Cook 134 not out and Bairstow 16 not out but the tourists were still 63 runs behind the hosts.
Lyon removed Bairstow and Ali in quick succession after lunch
After lunch, Bairstow and Cook returned looking to build on their partnership until Nathan Lyon came onto to bowl and he removed Bairstow for 22.
Moeen Ali, looking for some runs, was next at the crease, and despite smashing some runs, Lyon removed Ali for 20.
Cook and Woakes put on a half-century partnership for the seventh wicket to give the tourists the lead at Tea
Cook though was the man for England as he passed the 150 mark and alongside Chris Woakes, the pair saw England finally take the lead as the tourists finished the afternoon session on 360-6.
Quick wickets after tea left England needing some lower order runs
After tea, Woakes and Cook's half century partnership was brought to end when Cummins removed Woakes for 26.
Not long after that, Hazlewood then removed Tom Curran for four to leave the tourists on 373-8.
Cook and Broad put on a century partnership with both passing milestones
Broad, though taking some confidence from his bowling effort, done really well with the bat as he stuck around for Cook to reach his fifth test double century.
Broad and Cook really frustrated the Aussies in the final session of the day as Broad brought up his own half century as the pair took their partnership to the 100 mark.
The Aussies though did finally break the partnership when Cummins removed Broad for a well-made 56.
Cook ended the day unbeaten as the tourists take a sizeable lead into the fourth day
James Anderson toughed it out for the remainder of the day as the tourists ended day three on 491-9 with a lead of 164 with Cook the star man for the tourists on the day ending on 244 not out looking to keep going for as long as he can tomorrow.