Friday's shocking 1-1 draw with Finland took Croatia out of pole position in Group I and dropped the Balkan nation to second place on 17 points, two behind Iceland and ahead of Ukraine only on goal difference.
Andriy Shevchenko's men host Croatia in Kyiv on Monday with a play-off spot on the line.
Croatia managed to beat Ukraine by a slim 1-0 score in March, but Dalic has a much more difficult task on his hands.
The 50-year-old coached United Arab Emirates side Al Ain for the past three seasons, helping the club to the 2016 Asian Champions League final.
Cacic was under intense pressure even before Friday's slip-up. Losses to Iceland and Turkey in June and September, respectively, put Croatia in a precarious position.
Luka Modric, the star midfielder and captain for the Vatreni, said the situation is "catastrophic." Mario Mandzukic struck a 57th-minute opener for Croatia, only for Finland to snatch a point with an equaliser in second-half stoppage time.
"Until our match with Iceland, everything was fine and now all of a sudden nothing is of any worth," Modric said afterward, according to the Independent's Ed Malyon. "That's the honest truth. The situation is catastrophic, but we still have a chance to make it right."
Iceland isn't expected to lose its grip on top spot, considering it hosts bottom-feeder Kosovo at home Monday. Croatia, meanwhile, needs answers quickly to avoid missing the World Cup for the second time in eight years.