Resuming from his overnight score of 137, Symonds added 25 runs to eclipse his previous best of 156 before running out of partners before lunch.
India were yet to get off the mark after having played out three overs before lunch was called.
Lee chipped in with a sparkling 59, his fourth test half-century, as he and Symonds piled on 117 for the eighth wicket on a flat pitch still heavily favouring the batsmen.
Lee fell just five runs short of his highest test score when he was trapped lbw by spinner Anil Kumble but his dismissal failed to stop the run feast.
Mitchell Johnson smacked four boundaries in a quickfire 28 from 30 balls before he holed out to Saurav Ganguly in the deep off Kumble.
The Indian skipper mopped up the final wicket, trapping Stuart Clark lbw for a duck, to finish with figures of four for 108 after left-arm seamer RP Singh had taken four wickets on the first day.
Australia won last week's first test in Melbourne by 337 runs and victory in Sydney would equal their own world record of 16 consecutive test wins.