Chaudhary blazed his way to 163 for the Tigers before they declared at 8-623 late on day three and with a lead of 232 at Sydney's Cricket Central.
Sam Konstas and Ryan Hicks then survived a tricky five overs as the Blues went to stumps at 0-9 and needing to fight to hang on for a draw.
After Caleb Jewell hit 102 for Tasmania on Sunday, Tim Ward and Chaudhary both reached triple figures on Monday in the Tigers' second-highest Shield score.
Ward was dominant through the offside against his native state for his first red-ball century of the Shield summer, but it was Chaudhary who made history.
Born in Delhi and having represented Punjab in one-day cricket, Chaudhary moved to Australia in 2020.
He became stuck in the country through the COVID-19 pandemic, and has since become a permanent resident while still being an Indian citizen.
The spin-bowling allrounder was spotted in Queensland club cricket by Tasmania's assistant coach James Hopes, who recommended him to Hobart for the BBL two summers ago.
Chaudhary then moved to Tasmania at the start of last summer, and got his chance in the one-day cup this season.
He took a five-wicket haul on Shield debut against Queensland last month, before having his way with the bat on Monday.
The 29-year-old hit five sixes in his 184-ball knock, going after Tanveer Sangha late in the day.
Four times Chaudhary slog-swept Sangha over the rope in the final hour of play, bringing up his 150 when he put the legspinner on the grandstand roof.
A handful of Indian players have previously featured in the Shield, with former Test allrounder Rusi Surti hitting a century and taking a hat-trick for Queensland in the 1970s.
Aside from Chaudhary, Ward was the other Tasmanian to impress on Monday with 119.
He was dominant through the offside, regularly cover driving, and brought up his ton when he hit Sangha to the boundary.
Bradley Hope (91) also threw away the chance to join Chaudhary, Ward and Jewell as centurions, when he cut a Sangha lop hop to point.
The Tigers entered the round last on the ladder, one of four teams with one win, two losses and a draw to start the season.
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