Lassic attended North Rockland High School, where he helped his team achieve three straight Section 1 Class A bowl wins. As a senior, he set county records with 1,719 rushing yards, 31 touchdowns and 194 scored points, while being named the New York State large-school Player of the Year, All-state and an USA Today’s High School All-American. He also had 31 receptions for 586 yards. He practiced track and baseball. He accepted a football scholarship from the University of Alabama. As a redshirt freshman, he was fifth on the team with 142 rushing yards on 33 carries. In the 32-16 win against Louisiana State University, he rushed for 58 yards on 8 carries. In the 37–14 win against the University of Southern Mississippi, he made 4 receptions for 80 yards, including a 54-yard catch. That season Alabama won its first SEC championship since 1981. The next year, he was slowed by a turf toe injury, rushing for 281 yards on 68 carries. He scored his first touchdown in the season opener 24–27 loss against the University of Southern Mississippi. As a junior, he was the backup running back behind Siran Stacy, posting 368 rushing yards and 2 touchdowns. He became a starter the following year, registering 905 rushing yards on 178 carries, 10 rushing touchdowns, 129 receiving yards and one receiving touchdown. That season, he was named the 1993 Sugar Bowl MVP after Alabama beat Miami 34-13 to win the National Championship. He finished his career 11th on the school's all-time rushing list.
Lassic was selected by the Dallas Cowboys in the fourth round of the 1993 NFL Draft. The team was forced to start him in the first two games after Emmitt Smith contract holdout extended into the regular season. He became the first rookie running back in league history to start a season opener for a defending Super Bowl champion, and the third rookie running back in franchise history to start a season opener. Although Smith signed in time for the third game against the Phoenix Cardinals, Lassic still started and led the team in rushing with 60 yards on 14 carries, while scoring his first 2 touchdowns during a 17–10 win. From that point his workload was significantly reduced, on the way to the Cowboys winning back-to-back Super Bowls. In 1993, he rushed 75 times for 269 yards and 3 touchdowns, but he suffered from the national pressure and the loyalty his teammates had towards Smith. Erik Williams criticized Lassic's running style, saying "He runs too fast. We have a flow and he isn't able to see it", Nate Newton said, "Lassic was a nice kid, but if we expected to get back to the Super Bowl, we needed Emmitt in the lineup". The next season, he was limited with a torn ligament he suffered in his right knee during a pre-season game, until being placed on the injured reserve list on December 6, 1994. In Cowboys lore, he is remembered for being the starting running back for the first two games of the 1993 season when Smith held out in a contract dispute.
Carolina Panthers
The Carolina Panthers selected him from the Cowboys roster in the 20th round of the 1995 NFL Expansion Draft. The team reached an injury settlement with him on August 27, after suffering from knee, hamstring and thigh problems during training camp.
San Francisco 49ers
On March 18, 1996, Lassic signed with the San Francisco 49ers and was waived on July 29.
