Boswellia carterii is a plant species belonging to the Burseraceae family. It grows up in trees or shrubs, and it is known for producing an aromatic resin commonly named frankincense or olibanum. This resin has been used in traditional medicine to treat various conditions such as inflammations, gastrointestinal disorders and traumatic injuries. Virtual screening and molecular docking are two in silico approaches used to predict potential interactions between ligands and the active site of a protein. These approaches are mainly used in natural product chemistry and pharmacology as a screening tool to select plant extracts or fractions for in vitro testing, as well as for the prediction of mechanisms of action. The aim of this research is the in silico and in vitro evaluations of the potential collagenase and elastase inhibitory activities of Boswellia carterii resin organic extracts (viz., methanol, n-hexane and ethyl acetate). The obtained results revealed that methanol and n-hexane exhibited the best collagenase inhibitory activity with values superior to 85%, whereas the methanol and ethyl acetate showed the highest elastase inhibition activity with inhibition values ranging between 40 and 60%. The molecular docking prediction confirmed the experimental results; moreover, the visualization of the ligand–protein interactions showed that the main compounds of the organic extracts may have mechanisms of action similar to the positive controls. Those findings are very promising and open new perspectives for the exploitation of Boswellia carterii resin as active agents for the development of anti-aging cosmeceuticals.