Costanzo commenced his Formula 5000 career racing a Lola T332 purchased in partnership with his brother-in-law and mechanic Marino Ciuffetelli, competing in the Rothmans series from 1977 to 1979 with sponsorship from Stock 84 Brandy. In 1981 he moved to race under the patronage of Porsche Cars Australia distributor Alan Hamilton.
His first CAMS Gold Star title was won in a conventional Lola T430. The 1981 title, which was the last Formula 5000 national-level championship ever held globally, was claimed in a McLaren M26 modified to incorporate ground effect by Tiga Race Cars. Costanzo then won the 1982 and 1983 Australian Drivers' Championships in a Tiga FA81 powered by a 1.6-litre four-cylinder Ford BDA engine. In 1984 he finished second to longtime rival John Bowe — whom Costanzo's backer Hamilton had also fielded — driving a similar Ralt RT4/85, before Hamilton withdrew from open-wheel racing after Costanzo's fourth-place finish at the 1984 Australian Grand Prix.
Despite contesting eleven Australian Grand Prix events between 1969 and 1984, Costanzo never achieved the win that eluded him. His best result was fourth place in 1980 and again in 1984, on both occasions the leading Australian resident behind international Formula One drivers. At the 1983 Australian Grand Prix he led from the start and built a gap over eventual winner Roberto Moreno before a differential failure ended his race on lap 25. Moreno later acknowledged that Costanzo would have been very difficult to overcome. Costanzo also held an outright lap record at Symmons Plains Raceway — a 50.16-second lap set in the F5000 Lola T430-Chevrolet on 23 March 1980 — which stood unchallenged until 12 February 2022 when Joey Mawson recorded a 48.5598-second lap in a Rogers AF01/V8 S5000.
Costanzo was a capable touring car driver but was generally overlooked by the top teams and never established himself full-time in the category. He finished fourth in the 1979 Hardie-Ferodo 1000 alongside Allan Grice. At the insistence of rival-turned-friend John Bowe — who rates Costanzo as the best Australian driver he raced against — Costanzo was drafted into the Volvo Dealer Team in 1986 for the Castrol 500 at Sandown and the James Hardie 1000 at Bathurst, driving a Volvo 240T but failing to finish either event. Bowe again arranged a drive for him with Dick Johnson Racing in 1988, where he drove a Ford Sierra RS500, finishing sixth at the Sandown 500 alongside John Smith. The car intended for Costanzo at the 1988 Tooheys 1000 at Bathurst ultimately finished second overall, but he was sidelined when both Dick Johnson's and Bowe's cars expired and those drivers took over the Smith/Costanzo entry.
In the late 1990s Costanzo made a brief comeback as lead driver for Ciuffetelli's factory-supported Maserati team in the Australian GT Production Car Championship, driving Maserati Ghiblis. The campaign was short-lived due to uncompetitive machinery against Porsche, Ferrari, and Mazda rivals; he finished tenth in 1997 and eighth in 1998.
In December 1984 Costanzo co-drove the Romano WE84 Cosworth in the Sandown 1000, part of the 1984 World Endurance Championship — the first FIA World Championship event held in Australia. Despite having never driven the car previously, he qualified it thirteenth overall and fastest in the Australian Cars class, almost two seconds quicker than car owner Bap Romano and only 0.4 seconds behind the Group C2 pole time. Brake and gearbox problems restricted the pair to 109 laps, leaving them unclassified. Costanzo's reputation for being hard on equipment was reinforced during the meeting when he broke four gearboxes in total, two during the race itself, having habitually changed from fifth directly to second gear through a hairpin.