Pucovski rode his luck for his 62 but his partnership of 100 with Marnus Labuschagne for the second wicket established a solid platform for the hosts as they look to break a 1-1 tie in the four-match series.
Labuschagne notched his ninth test half century to finish the day on 67 not out with Steve Smith unbeaten on 31 at the other end in a potentially ominous stand between Australia's two best batsmen.
"Nice to spend a bit of time out there," said Smith, who had scored only 10 runs in his four previous innings in the series.
"Nice partnership building with Marnus and hopefully we can bat for a while tomorrow."
Australia's previous best tally in the series was their 200 in a losing effort in the second test in Melbourne and they overhauled their opening partnership for Sydney in an effort to improve matters.
David Warner returned after missing the first two tests with a groin injury but his stay at the crease was brief and he departed for five swatting at a Mohammed Siraj delivery that flew to Cheteshwar Pujara in the slips.
The rain descended soon afterwards with only seven overs completed and it was four hours before the skies cleared and Pucovski was able to return to the crease.
The 22-year-old was twice dropped behind the stumps by India wicketkeeper Rishabh Pant and survived a run-out scare but made the most of his reprieves to reach the half century with his fourth boundary.
His innings ended when he was trapped lbw to give Indian seamer Navdeep Saini his first test wicket as he also made his debut as a replacement for the injured Umesh Yadav.
"I was just absolutely loving it, it was probably my favourite day of cricket yet," Pucovski said.
"I'm really hopeful that with what we have, and the batting we have to come, we can build a pretty big total."
Siraj said his team mates would have to be patient in absence of any help from the wicket.
"It was a flat wicket, great to bat on," said the 26-year-old who made his test debut in the previous match in Melbourne.
"Our target will be to build pressure by hitting the same area and bowling dot balls after dot balls. We saw some turns towards the end, let's see how it pans out tomorrow."
After a second outbreak of COVID-19 in Sydney, the match was being played in front of a reduced attendance of around 10,000 fans with the wearing of masks compulsory for fans except when they are eating or drinking.
The fourth test is scheduled to start on January 15 in Brisbane but reports persist that India might refuse to head north because of the quarantine measures they will face in Queensland.
