They now play Glentoran at the same venue tomorrow.
In the earlier group game at Falkirk Stadium, Ukranian second seeds FC Zhilstroy confirmed themselves as the main danger to Eddie Wolecki Black's side with an identical scoreline against the Northern Irish champions. The two top teams meet on Thursday.
There was little evidence that the first half would end with City three goals ahead when it was still 0-0 after 25 minutes.
Although the Scottish champions were clearly the better side, the Nove Zamky defence, and particularly goalkeeper Annette Ndom and central defender Olga Tshepiso, looked more than capable of keeping the score down.
Ndom proved her worth with a save from City captain Leanne Ross in the ninth minute as the home side searched for an early goal.
Fiona Brown, sent clear by a clever pass from Suzanne Lappin, fired the ball over the bar and then Lappin's header from a Ross free-kick went wide.
A notable feature of the early play was the dominating performance of Irish international Denise O'Sullivan in midfield, with most of the promising City moves flowing through her.
But when the deadlock was finally broken with two goals in two minutes, both came from set plays.
The first saw Ross take a corner from the right and although Ndom made a fine one-handed save from Lappin's downward header, central defender Cheryl McCulloch swept the loose ball past the keeper.
The second goal was again from a Ross corner, this time from the left, and again scored by a central defender - a near-post glancing header from Eilish McSorley.
These were sore blows for the Slovakians after their stout early defending, and four minutes later they lost a third.
Abbi Grant's low cross from the left was met first time by Lappin, who steered the ball past Ndom.
Given the way the game had started, 3-0 was a rewarding interval outcome for Wolecki's side, but seconds before French referee Florence Guillemin blew the whistle Lappin nearly made it four, again from a troublesome Ross corner, but the Novy Zamky keeper pulled off a terrific save.
Predictably, the fitter home side started to assert themselves even more strongly in the second half. Brown missed two chances, and Ross another, before the fourth goal arrived in 61 minutes. Brown was the provider and O'Sullivan lashed a fierce shot past Ndom.
City eased down in the last half hour but still went on to equal the earlier Zhilstroy score deep into injury time.
The scorer was again McSorley, who netted direct from a free-kick 22 yards out.