Goalkeeper Danijel Subasic was the hero, saving three spot-kicks after the match finished 1-1 following extra time.
"Croatia in a crazy drama broke Denmark with penalties and progressed to the quarter-finals, Subasic hero of the match!" said the Tportal website.
"Subasic, who saved three penalties, was a hero, while Ivan Rakitic scored the decisive penalty," said the Vecernji List daily paper.
Fans watched the match live on giant screens throughout the country and celebrated the 3-2 win on penalties.
"Subasic saved us! He was an absolute hero of this tough match!" Fran Peric, 20, who watched the match in Zagreb, told AFP.
Fans, many of whom were dressed in Croatia's red-and-white shirts, were singing patriotic songs, lighting flares and waving Croatian flags.
"Subasic defends three penalties and leads 'Fiery Ones'," said the Sportske Novosti paper.
Croatia can match the achievements of the 1998 team, who reached the semi-finals, if they beat Russia in Sochi on Saturday.