Borzacchini was born in Terni in the Umbria region of Italy. His given name, Baconin, derived from that of Russian revolutionary anarchist Mikhail Bakunin, whom his parents had admired. At fourteen he began working in a garage, training as a mechanic. After serving in army artillery during World War I, he took up motorcycle competition before turning to automobile racing in 1926.
His first competitive successes came in hillclimbing. Over 1926 and 1927 he won six Italian hillclimb events driving a Salmson, and took class victories at the Targa Florio in both 1926 and 1927 in the 1100cc category, beating a fellow emerging talent, Luigi Fagioli, on both occasions. Victory at the Etna Cup at Catania and several other Italian events followed. These results brought him to the attention of the Maserati racing team.
Joining Maserati, he won the 1927 Terni-Passo della Somma and the Coppa della Collina Pistoiese. On 28 September 1929, driving a Maserati V4, he set a new flying 10-kilometre land speed record of 246.069 km/h (152.900 mph). That same year he finished second at both the Circuito di Alessandria and the Tripoli Grand Prix.
Maserati entered the 1930 Indianapolis 500 with Borzacchini as their driver. The car featured a radical swept-back aerodynamic front end, unusual for the era, and drew attention as a potential contender. Magneto problems eliminated him after only seven laps. He returned to European competition and won the 1930 Tripoli Grand Prix, adding another hillclimb victory at the Pontedecimo-Giovi event in Genoa.
In 1930, Borzacchini was pressured by the fascist government to race under the Italian name Mario Umberto rather than Baconin, effectively erasing a name his parents had chosen in tribute to revolutionary politics.
In 1931 Borzacchini signed to drive Alfa Romeos for Scuderia Ferrari, becoming a teammate and close friend of Tazio Nuvolari. He won the Circuito di Avellino but was repeatedly overshadowed by Nuvolari's brilliance, finishing second at the Targa Florio, the Grand Prix of Monza, and the Italian, Belgian, and French Grands Prix.
In 1932, driving the new Alfa Romeo P3, he finished second at the French Grand Prix and the Coppa Ciano, and third at the Großer Preis von Deutschland. His outstanding result of the season came at the Mille Miglia, which he won co-driving with Amadeo Bignami — a gruelling road race over public highways that ranked among the most demanding events in European motorsport. He ended the 1932 season as runner-up to Nuvolari in the European Drivers' Championship.
When Alfa Romeo withdrew from works racing after 1932 and sold its cars to Enzo Ferrari, Borzacchini returned to Maserati, joining teammates Luigi Fagioli and Giuseppe Campari. He opened 1933 with a second place behind Nuvolari at the Grand Prix of Tunisia, followed by second at Monaco behind Achille Varzi and third at the Avusrennen. At the Targa Florio he set the fastest lap before being forced out by accident.
On 10 September 1933, the Autodromo Nazionale Monza was the scene of one of the most catastrophic afternoons in pre-war racing. The Italian Grand Prix was run in the morning without serious incident, won by Luigi Fagioli. An additional afternoon event, the Monza Grand Prix run on the banked oval, was staged to draw larger crowds. On a track dampened by light drizzle, Borzacchini was racing for the lead with his Maserati teammate Campari when Campari's car slid on a patch of oil, flew over the top of the banking, and crashed fatally into the trees. Borzacchini, unable to avoid the same oil patch, spun into the infield and rolled. He was taken to hospital, where he died of his injuries later that day. A third driver, Polish count Stanislas Czaykowski, died when his car caught fire at the same location after a blown engine later in the restarted race.
Borzacchini competed in more than one hundred events over his career. Though his total of victories was smaller than those of the era's dominant figures, he was widely respected and loved in Italy. The Autodromo Mario Umberto Borzacchini at Magione in the Province of Perugia, not far from his native Terni, was named in his honor. He is buried in the local cemetery at Terni.