Ghana's qualification had been confirmed by FIFA ahead of the game, with their four points from victories over Panama and a draw with England enough to guarantee their place among the eight best third-placed teams regardless of the result.
Petar Sučić gave Croatia the lead on 31 minutes. Derrick Luckassen equalised on 73 to give Ghana hope of finishing second, but Vlašić struck on 83 to settle the game and send Croatia above them on the final standings.
Forward Brandon Thomas-Asante acknowledged the frustration of falling short of their group target despite the overall positive outcome.
"There's a degree of disappointment in camp," he said. "We knew that there was the opportunity to win the group. That was our intention, our ambition, and there was the possibility. So, of course, it was disappointing. We haven't done that, but we have to take the positive and we're into the next round. We want to go as far as we can in the tournament."
Ghana finish third in the group behind England on seven points and Croatia on six. They will face the winner of Group K in the round of 32 and make the knockout stage of the World Cup for the first time since their quarter-final run in 2010.
Croatia top the group having recovered from an opening 4-2 defeat to England to win their remaining two matches. England finish first after their 2-0 win over Panama in New Jersey, with Panama eliminated bottom of the group.