A native of Passaic, New Jersey, Malavasi entered the U.S. Military Academy, starting at offensive guard for the Black Knight football team. Under head coach Earl Blaik and line coach Vince Lombardi, Malavasi played two years, with Blaik rating him as the greatest line prospect during his tenure at the Academy. That potential disappeared when Malavasi was one of 90 cadets who left in the wake of a cheating scandal in August 1951.
Malavasi left to attend Mississippi State University, earning a degree in engineering while serving as an assistant under head coach Murray Warmath in 1952 and also receiving an Army ROTC commission. In 1953, he tried out and was released by the NFL Philadelphia Eagles, then served as line coach with the Fort Belvoir army unit for two years beginning in 1954.
Upon his release in 1956, Malavasi accepted an assistant's role with the University of Minnesota, spending two seasons with the Golden Gophers until taking a similar position with Memphis State University in 1958. Three years in Tennessee led to a single year with the Wake Forest University Demon Deacons in 1961, followed by his first professional position the following year: personnel director of the American Football League's Denver Broncos.
In 1963, Malavasi also took on the duties of defensive line coach for the Broncos, and shifted to the offensive line prior to the start of the 1966 season. After Denver dropped their first two games, head coach Mac Speedie was fired, with Malavasi chosen to replace him. In the final 12 games, Malavasi managed just four wins and was let go in favor of Lou Saban at the conclusion of the season.
For the next two years, Malavasi coached on the defensive side of the ball with the Canadian Football League's Hamilton Tiger-Cats, then accepted the defensive line assistant's position with the Buffalo Bills on February 3, 1969. Just over two years later, Malavasi left to work for the Oakland Raiders, but resigned after just two years, citing the frustration with the team keeping him from advancing his career.
Malavasi's resignation caused controversy when Raiders' head coach John Madden accused another team of tampering with Malavasi's services. While he denied the charges, Malavasi was hired on June 5, 1973 as a defensive assistant with the Los Angeles Rams. Over the next five years, Malavasi rose to become defensive coordinator, and was under consideration for head coaching positions with both the Eagles and Chicago Bears. He also was looked at as head coach for the Rams after the departure of Chuck Knox following the 1977 NFL season, but instead stayed as an assistant under new coach George Allen.
However, after just two exhibition games, Allen was fired and Malavasi became the team's head coach, then led the team to their sixth straight NFC West Title with 12-4 record, reaching the NFC Championship game. The following year, the team barely finished above .500 with a 9-7 mark, but won the weak NFC Western Division. In the playoffs, the team upset the Cowboys, then shut down the upstart Tampa Bay Buccaneers 9-0 in the conference title game to give the franchise its first Super Bowl berth ever. After entering the fourth quarter with a lead over the heavily-favored Pittsburgh Steelers, the Rams' upset bid died when they dropped a 31-19 decision in Super Bowl XIV.
An 11-5 mark in 1980 was a two-game improvement over 1979, but it was only good enough for second place behind the Atlanta Falcons in the NFC West, and the Rams fell in the wild-card round to the Cowboys. The following year, the rise of the San Francisco 49ers ended the Rams' reign and saddled them with a 6–10 record. Following a 2–7 mark during the strike-shortened 1982 NFL season, the worst record in the 14-team NFC, Malavasi was dismissed on January 4, 1983.
Malavasi served as a consultant and unofficial spokesman for the proposed International Football League, a league that served the dual purpose of introducing professional football abroad as well as serving as an antagonist to the United States Football League (Malavasi vowed that the league would hold its draft and opening kickoff one week before the USFL's respective dates, no matter what the USFL did). The IFL folded without any teams organizing; after the league's failure, Malavasi resurfaced as an assistant with the USFL's Oakland Invaders in early 1984. He left that position just after the start of the season to become defensive coordinator of the league's Los Angeles Express, but the financial woes of the team ended his brief tenure in the final weeks of the season.
Malavasi would never again be on the sidelines for an NFL team but he did coach the first ever Australian national American Football team in 1987. The Australian Kookaburras did a week-long training camp at Dominguez Hills in California before heading off for a tour of Europe where they played three games. Malavasi was assisted with the team by Johnny Johnson. Malavasi died of a sudden heart attack on December 15, 1987, aged 57, less than two months after returning from that tour.