Tendulkar, still chasing a 100th international hundred, looked to settle his side but was dismissed for 32 when Siddle, with the first ball of his second spell, had him caught in the gully off a thick edge.
Mahendra Dhoni and Ravichandran Ashwin attempted to launch a counter-attack after tea but fell to Pattinson and Siddle respectively as wickets continued to fall.
Australia clinched victory in style when David Warner caught Umesh Yadav with a dramatic catch on the boundary.
"For me it's a dream come true to play for Australia and to be out there on Boxing Day, and against India with that batting line-up," 21-year-old Pattinson said.
"I think I wasn't even born when Sachin started playing cricket, so to go out there and bowl against him, and have some success against their batsmen and get a win is even better."
Australia v India
1st Test, Melbourne
2nd Test, Sydney
3rd Test, Perth
4th Test, Adelaide
1 & 3 Feb:
5 Feb-2 Mar:
One-day series - Australia, India, Sri Lanka
Australia captain Michael Clarke said his side had to maintain their intensity in the second Test, which starts in Sydney on 3 January.
"India are a fantastic team and it's great to get a win but we know they'll come back hard at us," he said.
"We deserve this win because of the amount of effort and time we've put into our preparation, but it means nothing come Sydney."
India captain Dhoni said: "You have to give credit to the Australian bowling attack because they bowled really well.
"We thought if we could get them out for 240 or 250-odd runs that's a very gettable score, but I felt 290-odd runs was also a score we should have achieved.
"The wicket was pretty good. It's not like there was too much wear and tear in the wicket. I think our batting line-up flopped in both the innings."
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