Rosmarinic acid is a hydroxycinnamic acid ester commonly found in the Boraginaceae and Lamiaceae plant families. It exhibits various biological activities, including antioxidant, anti-inflammatory, antibacterial, antiallergic, and antiviral properties. Rosmarinic acid is used as a food and cosmetic ingredient, and several pharmaceutical applications have been suggested as well. Rosmarinic acid is currently produced by extraction from plants or chemical synthesis; however, due to limited availability of the plant sources and the complexity of the chemical synthesis method, there is an increasing interest in producing this compound by microbial fermentation. In this study, we aimed to produce rosmarinic acid by engineered baker’s yeast Saccharomyces cerevisiae. Multiple biosynthetic pathway variants, carrying only plant genes or a combination of plant and Escherichia coli genes, were implemented using a full factorial design of experiment. Through analysis of variances, the effect of each enzyme variant (factors), together with possible interactions between these factors, was assessed. The best pathway variant produced 2.95 ± 0.08 mg/L rosmarinic acid in mineral medium with glucose as the sole carbon source. Increasing the copy number of rosmarinic acid biosynthetic genes increased the titer to 5.93 ± 0.06 mg/L. The study shows the feasibility of producing rosmarinic acid by yeast fermentation.