Starc was closing in on a maiden first-class century during an unbroken 189-run stand with Sean Abbott - who had just reached his first century - when Nevill waved them in so that New South Wales could have a short session with the ball late on the third day.
As Starc headed towards the New South Wales changing area he hurled his bat and gloves in frustration with team-mate Nathan Lyon admitting after the match that he would have liked one more over to try and reach the century.
Starc has a career-best of 99, which came against India in Mohali, and he also came close to a hundred against Pakistan at the MCG in 2016 when he struck 84.
"We have had a bit of a chat about it," Nevill said after New South Wales wrapped up an astonishing victory having been bowled out for 64 on the opening day. "I did apologise to him as well, I'd love for him to get that milestone but the most important thing was us having a crack at them for 45 minutes or whatever it was last night. To get two key wickets last night made the job today that much easier.
"He's obviously frustrated because he was batting really well and close to getting a hundred. They'll be another opportunity for him, no doubt, if he keeps batting that way."
Speaking from the viewpoint of a fellow lower-order batsman, Lyon sympathised with Starc but insisted he would have wanted what is best for the team
"I can tell you one thing, as one of my best mates, if we get close to a hundred like that they don't come around very often," he said. "It's unfortunate he didn't get the opportunity, he would have liked one more to give it a crack. He supported Sabba [Sean Abbott] extremely well, but Starcy is the least selfish team man I've ever met, he's always about the team and always wanting the team to win. He's alright."
Nevill's declaration worked perfectly as Trent Copeland claimed two wickets meaning New South Wales needed just eight on the final day. After a wicketless opening session, Lyon struck straight after lunch and victory was secured in the last session.
It meant that New South Wales' first innings 64 became the second lowest first-innings total to be on the winning side in the Sheffield Shield after South Australia's 55 all out in 2010, which also came against Tasmania, although South Australia did with a match in 1968 after declaring at 7 for 40.
"I was thinking about it on the field today, I can't think of too many better victories I've been involved in than that one," Nevill said. "To be in that sort of position and be able to fight back to dominate the next three days and actually win the game by a significant margin is an incredible performance."
Andrew McGlashan is a deputy editor at ESPNcricinfo