The medicinal plants of the Asteraceae family are a valuable source of bioactive secondary metabolites, including polyphenols, phenolic acids, flavonoids, acetylenes, sesquiterpene lactones, triterpenes, etc. Under stressful conditions, the plants develop these secondary substances to carry out physiological tasks in plant cells. Secondary Asteraceae metabolites that are of the greatest interest to consumers are artemisinin (an anti-malarial drug from Artemisia annua L.—sweet wormwood), steviol glycosides (an intense sweetener from Stevia rebaudiana Bert.—stevia), caffeic acid derivatives (with a broad spectrum of biological activities synthesized from Echinacea purpurea (L.) Moench—echinacea and Cichorium intybus L.—chicory), helenalin and dihydrohelenalin (anti-inflammatory drug from Arnica montana L.—mountain arnica), parthenolide (“medieval aspirin” from Tanacetum parthenium (L.) Sch.Bip.—feverfew), and silymarin (liver-protective medicine from Silybum marianum (L.) Gaertn.—milk thistle). The necessity to enhance secondary metabolite synthesis has arisen due to the widespread use of these metabolites in numerous industrial sectors. Elicitation is an effective strategy to enhance the production of secondary metabolites in in vitro cultures. Suitable technological platforms for the production of phytochemicals are cell suspension, shoots, and hairy root cultures. Numerous reports describe an enhanced accumulation of desired metabolites after the application of various abiotic and biotic elicitors. Elicitors induce transcriptional changes in biosynthetic genes, leading to the metabolic reprogramming of secondary metabolism and clarifying the mechanism of the synthesis of bioactive compounds. This review summarizes biotechnological investigations concerning the biosynthesis of medicinally essential metabolites in plants of the Asteraceae family after various elicitor treatments.