The TR3 is an evolution of the earlier TR2 model, featuring greater power and improved braking. Though an open two-seater, optional extras included an occasional rear seat and a bolt-on steel hard top. The car’s 1,991 cc engine initially produced 95 bhp, later increased to 100 bhp at 5000 rpm with the addition of a "high port" cylinder head and enlarged manifold. Front brakes were changed from drums to discs in 1956, a first for a British series production car.
The TR3 was updated in 1957 to become the “TR3A”, though this name was not used officially by Triumph. Changes included a new full-width front grille, exterior door handles, and a lockable boot handle. The previously optional full tool kit became standard equipment, and the TR3A carried over the front disc brakes introduced on later TR3s. In 1959, further modifications were made, including raised stampings under the bonnet and boot hinges, a redesigned rear floor section, and a windscreen attached with bolts instead of Dzus fasteners. A 2138 cc engine and 60-spoke wire wheels were available as options that year.
The final version of the TR3, unofficially known as the "TR3B", was produced in 1962. This short production run was a response to dealer concerns that the public might not accept the TR4, which began production in 1961. Most TR3Bs appeared identical to late US-model TR3As, with wider headlamp rims, a wider grille, and door handles. Two series of the TR3B were made: 530 cars with commission numbers preceded by TSF, and 2,804 cars with commission numbers preceded by TCF. The TCF series featured a fully synchronised transmission and a 2138 cc engine producing 105 hp and 172 N⋅m of torque.
The TR3 was campaigned extensively in motorsport. After the 1955 Le Mans disaster, Triumph competition manager Ken Richardson had steel hard tops bolted to 100 TR3s to homologate them as “grand touring” coupés, allowing them to compete in French motorsports. The TR3 competed in rallies including the RAC, Monte Carlo, Circuit of Ireland, Alpine, and Liege-Rome-Liege, achieving numerous outright, team, and class victories, including six "Coupes des Alpes" awards. At the 1959 24 Hours of Le Mans, three modified TR3s, designated ‘TR3S’ models, were entered, one of which was running as high as seventh place overall before retiring due to mechanical issues.
British auto magazine The Motor tested a hardtop TR3 with overdrive in 1956, recording a top speed of 105.3 mph and 0–60 mph acceleration in 10.8 seconds. Road & Track magazine published an article in June 1977 titled "Driving Impressions: TR3A & TR250", reporting a 0–60 mph time of 12.0 seconds for the TR3A, with a power output of 100 bhp at 4800 rpm.
Prior to the TR3, Giovanni Michelotti was invited by Triumph managing director Alick Dick to produce a concept car. Michelotti responded with the Triumph TR3 Speciale, built by Vignale on an unmodified TR3 chassis, incorporating styling cues from contemporary American practice, including tailfins and a full-width grille. The TR3 Speciale debuted at the Geneva International Motor Show in March 1957, but Triumph deemed it too expensive for production.