The former keeper-batter, known for his swashbuckling style, also expressed confidence in Marnus Labuschagne bouncing back to form.
After Bumrah scythed through the Australian batting line-up, taking eight wickets including a five-for in the first innings at Perth, Gilchrist advised the hosts to "weather the storm".
Steve Smith and Labuschagne, the team's two batting mainstays, flopped in he series opener and that did not help Australia's cause.
"Marnus had the onus to do that and tried at good effect to nullify 50-odd deliveries. So, if you face an average of 50 deliveries every Test innings, you're probably going to be on the right side of the ledger more than you're on the wrong side of it," Gilchrist was quoted as saying by Nine's Wide World of Sports.
"He just couldn't find a way to score and maybe that's what as a collective will look to try and do.
"It will bring with it risk, for sure, but it is a risk for reward."
Queenslander Labuschagne has been struggling for runs in recent months and now finds himself under immense pressure to retain his place in the side.
Gilchrist attributed Labuschagne's poor run of form to conditions and the quality of bowlers in the opposition camp, and didn't see it as a technical issue.
"I'm sure the people around him are already doing this but he needs to be reminded that he is a class act," Gilchrist said of Labuschagne.
"He's got a game that has been able to tolerate and withstand everything that the cricketing world has presented, and he's dominated for a number of years."
The former star added, "Most players that have played long enough will have been in this situation at some point in time and it's about not letting any self-doubt creep in and understanding you don't lose that skill — you don't lose that edge — certainly not in a short space of time."
Gilchrist's urged Labuschagne to "trust your training, get out there, see the ball and play it."
The second Test, a day/night affair, will begin on Friday at Adelaide Oval.
India lead the five-match series following their 295-run victory in the opener at Perth last week.
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