The Italians, who said after their 9-8 loss to the United States in the semi-finals that they had given their opponents too many chances, were miserly from the start in the bronze medal clash.
The world champions did well to limit Britain's scoring opportunities and steal points at every opportunity to build a 3-2 lead heading into the sixth end.
In that end, Britain's Bruce Mouat and Jennifer Dodds set up three stones in scoring position, but under tremendous pressure, Constantini threaded her final shot through a narrow gap and onto the button to clinch a crucial point.
Italy controlled the closing stages, restricting their opponents to one point in the penultimate end before scoring one more with their powerplay to secure victory, sparking massive celebrations from the partisan crowd at the Cortina Curling Olympic Stadium.
Constantini and Mosaner had beaten the British duo three times previously, once at the Beijing Games in 2022 and twice at last year's World Championships, including in the final, but they lost 9-6 when the pairs met in the round-robin stage in Cortina.
For Britain's Mouat and Dodds, it marked another crushing fourth-place finish after also missing out on a medal in Beijing, though this disappointment might sting even more in light of their impressive round-robin performance.