The goal drained the life out of Hampden Park and Scotland's fortunes did not look like improving as Croatia stroked passes around the pitch, but just as the World Cup finalists seemed likely to turn the screw further, Steve Clarke's side struck back.
Deprived of Billy Gilmour, who failed a COVID-19 test in the buildup, Scotland seemed lacking in midfield and Croatia bossed the ball, dominating possession for large periods before deservedly taking the lead after 17 minutes as Vlasic scored his sixth goal for his country.
However, Scotland drew level just before the break when a poor clearance by Croatia fell to the feet of Callum McGregor, who composed himself and struck a perfect low shot past Croatia keeper Dominik Livakovic to make it 1-1 at half-time.
Croatia retook the lead in the 62nd minute when Mateo Kovacic laid the ball off at the top of the penalty area to Modric, who curled a lovely shot with the outside of his right foot outside of the reach of Scotland's David Marshall.
"After tonight's performance, we'd have been able to walk off the pitch with our heads high even if we had lost," said Modric, now both Croatia's youngest and oldest scorer at the Euros.
"The goal I scored means a lot to me, I am glad it helped us win but the most important thing is that the whole team looked the part."
Modric then teed up Perisic for Croatia's third, with a flicked-on header doubling his team's advantage and ending Scotland's hopes of advancing to the round of 16.
On a down note for the losers, Scotland's dreadful run of never making the knockouts at a major tournament in 11 attempts continued.
"I think they just had too much quality in the end, you see the second goal and Modric with the outside of his foot puts it in the top corner," said Scotland goalscorer McGregor. "That's what you're up against at this level. It's a tough lesson."
Information from Reuters was used in this report.