Goals from Cheryl McCulloch, Eilish McSorley (two), Suzanne Lappin and Denise O'Sullivan overcame initial stubborn resistance by the Slovaks.
The other group match, played at Falkirk Stadium, also resulted in a 5-0 win for Ukrainian champions FC Zhilstroy against Glentoran.
If, as expected, the two group leaders win tonight - Zhilstroy play Nove Zamky - the match to decide which team reaches the last 32 will be played at the Excelsior Stadium on Thursday. Only the eight group winners are guaranteed to reach the knockout stages, although there will be places for the best two runners-up.
Wolecki Black watched the Zhilstroy match before preparing his side for their evening game on Saturday. He said: "There will be a lot of changes to our side for the Glentoran game because it's impossible to go from Saturday to Monday with one day's rest and put out the same group of players."
Tonight's game is also at the Excelsior, with a 7.45pm kick-off.