Finalists in 2018 and semi-finalists four years later, Croatia carried the greater threat in the first half and went close to taking the lead on 17 minutes, when Vlasic's long-range drive grazed the outside of goalkeeper Benjamin Asare's right-hand post.
But Dalic's team took a richly deserved lead when Inter Milan midfielder Sucic collected Mateo Kovacic's pass before arrowing a low, right-footed drive through Luckassen's legs and into the bottom corner.
Manchester City winger Antoine Semenyo dragged a low shot narrowly wide of the far post late in the first half, but it would have been a scarcely deserved equaliser for Carlos Quieroz's side, who failed to muster a first-half shot on target for the third game in succession.
However, the Black Stars appeared revitalised by the half-time introduction of Leicester City winger Abdul Fatawu, who flashed a powerful shot narrowly over the bar not long after the restart.
Ghana were rewarded for their improved second-half showing when Luckassen headed home Ernest Nuamah's delivery – a goal which was awarded by the video assistant referee following a near four-minute check for offside.
Kwasi Sibo was in an offside position from Nuamah's cross, but on-field official Drew Fischer eventually informed the crowd that the Ghana number eight was not interfering with play when the ball was struck and gave the goal.
The momentum appeared to be with Ghana as the game entered the final 10 minutes, but Mario Pasalic forced Asare into a magnificent one-handed save before Vlasic nodded home Luka Modric's corner to earn Croatia their second Group L victory.