“Very happy. We were put under pressure in the first innings but the way we responded was great,” said stand-in skipper Bumrah.
“We were really well prepared. I told everyone to keep faith in your ability.”
It was an astonishing turnaround for India, who arrived in Australia on the back of a humiliating 3-0 home series defeat to New Zealand.
Given little hope of defending the Border-Gavaskar Trophy they have held since 2017, they now head to the second day-night Test in Adelaide next week with a big psychological advantage and boosted by the return of skipper Rohit Sharma.
The emphatic victory was only India’s second in Perth and first since their triumph at the WACA Ground back in 2008.
“Fairly disappointing. Prep was good, all firing, fair bit to look at as lots didn’t go right,” said Australia captain Pat Cummins.
“You want to get back on the horse but we will take a couple of days' rest and get training in Adelaide.
“We didn’t give ourselves a chance in a few different facets, like the finish on day one with the bat.
“There will be lots of conversations on what we can do better in same conditions.”
Under Bumrah’s makeshift captaincy, India dominated an ageing side which is now the one facing scrutiny after failing to fire.
Out for a dismal 104 in reply to India’s first-innings 150, the hosts' bowling attack then had no reply to a brilliant 161 from impressive young opener Yashasvi Jaiswal and an unbeaten 100 from Virat Kohli.
The superstar batsman Kohli came into the five-match series under pressure.
“Jaiswal’s best Test innings so far, he left the ball well,” Bumrah said of the 22-year-old.
“I didn’t see Virat out of form - difficult to judge that on difficult pitches. But he was good in the nets.”
Australia’s batting line-up, which has long relied on one large score or partnership to pull them out of sticky situations, was mostly outsmarted.
With India setting them a huge target, they again capitulated on Monday.
After imploding in the final 30 minutes of play on Sunday, they resumed on 12-3 with Usman Khawaja on three and Steve Smith yet to score.
Khawaja added just one when he mistimed a pullshot off Siraj and got a big top edge.
It was caught by backtracking wicketkeeper Rishabh Pant, who became the most expensive player in Indian Premier League history Sunday at the Twenty20 tournament’s lucrative auction.
Head survived a loud lbw shout on seven, a review found it was missing leg stump and he battled on smacking seven boundaries for his 17th Test half-century.
At the other end, Smith was felled by a Harshit Rana bouncer that slammed into his midriff, needing a spell lying on the ground to recover.
He was able to get up and play on, and like Head came through an lbw review on 12.
But their partnership was ended by Siraj, with Pant taking another neat catch after Smith edged a delivery he had to play on 17.
Head teamed up with Marsh in an 82-run stand to temporarily raise home hopes before Bumrah again worked his magic.
Seemingly destined for a century, the largely untroubled Head feathered to Pant with Bumrah’s loud double fist-pump showing how ecstatic he was at the breakthrough.
Allrounder Marsh kept the scoreboard ticking over with Alex Carey but fell for 47, dragging a wide Nitish Kumar Reddy delivery onto the stumps.
Mitchell Starc, top-scorer in the first innings, was out for 12 on the cusp of tea before the tail was wrapped up early in the final session.
