Tansy (Tanacetum vulgare) is a common plant used in folk medicine for digestive problems, fevers, and migraines; against parasites; and as an insect repellent. The active substances in essential oil are responsible for its antimicrobial and antioxidant activity. Thus, tansy essential oil (TO) was added to alginate films to fabricate materials with antioxidant and antibacterial properties for food packaging. Sodium alginate films with glycerol and TO were tested in terms of structure, mechanical, thermal, antioxidant, and antibacterial properties. The structure of the films was examined using SEM and an ATR-FTIR spectrophotometer. The addition of TO to the alginate film significantly changed the films’ microstructure, making them rougher and porous. A low-intensity band at 1739 cm−1, indicative of the presence of TO, appeared in all spectra of alginate films with TO. Moreover, the studies revealed that essential oil acted as a plasticizer, slightly reducing tensile strength from about 7 MPa to 5 MPa and increasing elongation at break from 52% to 56% for the sample with 2% TO. The alginate films enriched in TO exhibited antioxidant properties (280 μmol Trolox/100 g of the sample with 2% TO) and antibacterial activity against Escherichia coli, Staphylococcus aureus, and Pseudomonas aeruginosa.