The unlikely pair combined for 55 and batted for more than an hour until stumps on day four at the MCG, with Australia 9-228.
Indian pace sensation Jasprit Bumrah (4-56) appeared to have Lyon (41no) caught in the slips in the day's final over, but the Australian spinner was walking off when he was called back due to a no-ball.
Lyon and Boland (10no) have faced 107 balls in their partnership to frustrate India's push for a win that would ensure they retain the Border-Gavaskar trophy for a record-extending fifth-straight time.
The match is set up for a day-five thriller, with India needing to pull off the highest successful run chase at the MCG, topping England's 332 in 1928.
A victory would also make it India's third-highest successful Test run chase - eclipsing even the 328 they famously scored at the Gabba to stun Australia in 2021.
India's only other run chases higher are 406 against the West Indies in 1976 and 387 against England in 2008.
Jasprit Bumrah ripped apart Australia's middle order in the fourth Test. (James Ross/AAP PHOTOS)
Bumrah (eight wickets for the Test) produced another classic spell to decimate Australia's top-order and give India a fighting chance of a special comeback.
He instigated a collapse of 4-11 in 21 balls after lunch, as Australia slumped from 2-80 to 6-91.
Marnus Labuschagne (70), who had a life on 46 thanks to the second of Yashasvi Jaiswal's three dropped catches, added to his 72 in the first innings to finish a calendar year without a Test century for the first time since 2018.
Pat Cummins' knock of 41 - following his first innings 49 - was equally important, with the captain batting sensibly with Labuschagne and the rest of the tail.
Cummins made more runs in this Test than struggling Allan Border medallist Mitch Marsh's tally of 73 for the entire series.
"I had a fair feeling lining him (Bumrah) up, and he was hot," Labuschagne said.
"I I just said to Pat when he came out ... Â I'll just take Bumrah and we can run on the other guys.
"Let's just make sure that I'm at the non-striker's end at the end of each over."
In a frantic three-over spell after lunch, Bumrah claimed the wickets of Travis Head (1), Marsh (0) and Alex Carey (2), giving him figures for that period of 3-4.
The dismissal of Head, a soft catch to square leg, was Bumrah's 200th Test wicket.
The 31-year-old reached the 200-wicket milestone with a better career average - 19.56 - than any bowler in history.
Bumrah has 29 of the 66 wickets India has taken during this Border-Gavaskar series at a measly average of 13.24.
Usman Khawaja couldn't capiitalise on his start, bowled by Mohammed Siraj for 21. (James Ross/AAP PHOTOS)
But he did tire dramatically late in the day after bowling 24 overs to go with his 29 in the first innings.
Before lunch, Bumrah exacted revenge on Sam Konstas for the teenage debutant's Boxing Day ramp-fest.
After writing himself into Boxing Day folklore with an audacious 60 on Thursday, Konstas went for eight in his second Test innings when Bumrah bowled him between bat and pad.
Veteran opener Usman Khawaja avoided falling victim to Bumrah for the sixth time in the series, but just when he looked set the left-hander was bowled by Mohammed Siraj for 21.
Siraj shushed the crowd, having been booed by Australian fans since giving Travis Head a send-off in Adelaide. The fiery quick ended the day with 3-66 after also taking the key scalps of Labuschagne and Steve Smith (13).
Nathan Lyon removed century-maker Nitish Kumar Reddy, finishing the innings with 3-96. (James Ross/AAP PHOTOS)
A brilliant maiden century from Nitish Kumar Reddy powered India past the follow-on target on Saturday, after the tourists were in trouble at 5-164 in reply to Australia's 474.
Reddy, just 21 and playing in his fourth Test, helped India add another 11 runs to their overnight score of 9-358.
""I feel so happy that my first century, my father is here ... it's a special moment for me," Reddy said.
"I held my team in a difficult situation to where we can fight back strong, so that was a special century for me and for my father as well."
Mitchell Starc appeared to show the effects of a tough campaign on Saturday. (James Ross/AAP PHOTOS)
