Ivy can be invasive and difficult to eradicate. Yes, you can certainly mow and then smother it. Or cut it back to the ground with a weed trimmer.  However, landscape fabric is no longer something that is recommended for use  as it can cause many more problems than it solves including water penenetration issues, soil compaction, and as it breaks down over time it can leach harmful chemicals and microplastics into the soil.  

Alternatives to using landscape fabric include placing a thick layer of a compostable material down and then covering with mulch.   For example, use a thick layer of  cardboard, then cover with 8 inches of organic matter like wood chips or bark. This method can reduce ivy stands by up to 80% after several months.  You could also try plywood, and then remove it after the ivy has been killed if the plywood does not break down quickly enough.  

Another possibiity is to spray the ivy with an herbicide such as Triclopyr or 2-4 D.  You could cut the ivy first and then apply the herbicide to the cut surface.  However you will want to be careful about your application to not get it on tree trunks or desirable plants.  Always read the label and follow instructions.  Herbicide is best used when plants are actively growing, so if the plant is starting to go dormant you would need to wait until new growth in the spring to use this method.  

You could also combine the methods, first using herbicide and then once the plants have mostly died, follow-up with a thick layer of cardboard and mulch.