The Ranji Trophy champions, who resumed at their overnight 150 for 4, were struggling at 179 for seven before they recovered through some resolute tail-end batting in the first session which was extended by half an hour to make up for lost time on the first day due to inclement weather.
At lunch Vinayak Samant, 19 after a stay of 92 minutes, and Nilesh Kulkarni (15 in 53 minutes) were holding fort for the hosts.
Replying to Australia's 302 for seven declared, Mumbai lost their first two wickets in the morning in only 14 balls, but recovered poise through the eighth wicket stand of 34 between Ajit Agarkar and Vinayak Samant which took their score past 200.
Agarkar, included in the first Test team yesterday, feasted on the spin bowling of Michael Clarke to make 37 and dominated the stand.
He was finally out giving a charge to the Aussie left arm spinner and managing only to sky a catch to McGrath who took a well-judged catch at mid off. Agarkar struck six fours in his 41-ball cameo.
Earlier, the Aussie pace trio of Glenn McGrath, Shane Watson and Brett Lee made early inroads into the home teams batting line up.
McGrath started the slide by trapping Amol Muzumdar, 50 last evening, leg before while shuffling across in his very first over of the morning after the batsman had added only two to his individual score.
It was the gangling Aussie pace bowler's third wicket of the innings and no Mumbai batsman played him with conviction.
Watson sent back Ramesh Powar by having the stocky batsman caught flicking at mid wicket and then Lee got his first wicket of the match by luring Mumbai skipper Sairaj Bahutule edge to wicketkeeper and Aussie captain Adam Gilchrist.
The Aussies took the second new ball after 86 overs, when the spinners failed to terminate the innings. Both McGrath and Lee targeted the tail-enders Vinayak Samant, who can bat a bit, and the lanky Nilesh Kulkarni for some short stuff aimed at the batsmens rib cage.
McGrath had some words to say to Kulkarni after being driven for an all-run four to the extra cover region while Lee struck the player on his knuckles with a snorter.
But the two batsmen not only stayed put but also added 34 runs for the unbroken ninth wicket before the interval.