Arnica montana L. is a medicinal plant with diverse biological activities commonly used in pharmacy and cosmetics. The attributes of A. montana are mainly related to the concentration and chemical composition of essential oils (EOs). Therefore, the objective of this study was to characterize the chemical composition of EOs derived from A. montana rhizomes and roots taking into account the age of the plants and to investigate the effect of the analyzed EOs on induction of apoptosis, necrosis, and autophagy in human glioblastoma multiforme T98G and anaplastic astrocytoma MOGGCCM cell lines. Rhizomes and roots of mountain arnica were harvested at the end of the third and fourth vegetation periods. The chemical composition of essential oils was determined with the GC–MS technique. Among the 37 components of the essential oil of A. montana, 2,5-dimethoxy-p-cymene (46.47%–60.31%), 2,6-diisopropylanisole (14.48%–23.10%), thymol methyl ether (5.31%–17.79%), p-methoxyheptanophenone (5.07%–9.65%), and α-isocomene (0.68%–2.87%), were detected in the rhizomes and roots of the three-year-old plants and in the rhizomes and roots of the four-year-old plants. The plant part (rhizome, root) and plant age can be determinants of the essential oil composition and, consequently, their biological activity. The induction of apoptosis (but not autophagy nor necrosis) at a level of 28.5%–32.3% is a promising result, for which 2,5-dimethoxy-p-cymene, 2,6-diisopropylanisole, thymol methyl ether, and p-methoxyheptanophenone are probably mainly responsible. The present study is the first report on the anticancer activities of essential oils from A. montana rhizomes and roots.