Born February 24, 2000, in St. Charles, Missouri, Dominick Puni comes from a family of athletes. His parents, Teu and Jessica, were both college athletes at Truman State University, where his father played football and his mother played volleyball. His brothers, Derrick and Devin, played football at Central Missouri, and his sister, Brianna, played basketball at Illinois State. Puni attended Francis Howell Central High School in Cottleville, Missouri, where he did not allow a single sack during his high school career.

Puni played college football at Central Missouri from 2018 to 2021, appearing in only one game in 2021 due to injury after the team's 2020 season was cancelled because of the COVID-19 pandemic. He transferred to the University of Kansas in 2022 and started all twenty-four games over two seasons, earning first-team All-Big 12 honors in 2023. During his senior season, Puni allowed zero sacks and zero quarterback hits over 342 pass-blocking snaps while posting a 90.4 pass-blocking grade. He was named a finalist for the Polynesian College Football Player of the Year Award and opted out of the 2023 Guaranteed Rate Bowl to prepare for the NFL draft.

The San Francisco 49ers selected Puni in the third round with the eighty-sixth overall pick in the 2024 NFL Draft after trading up with the Philadelphia Eagles. When injuries struck the offensive line during training camp, Puni won the starting right guard position despite never having played the position in college. Strengths include powerful run-blocking with excellent body control and grip strength, strong pass protection with a firm anchor, and exceptional athletic ability for his size, demonstrated by his 5.35-second forty-yard dash and thirty-inch vertical jump at the NFL Combine.

Puni's standout rookie performances included earning the team's highest Pro Football Focus grade (83.4) against Tampa Bay in Week 10, posting an 87.8 grade against Seattle in Week 6 while helping protect quarterback Brock Purdy on twenty-eight pass attempts with zero sacks allowed as the offense rushed for 228 yards, and earning an 85.8 grade against New England in Week 4.

Puni started all seventeen games in 2024, playing 100 percent of the team's 1,078 offensive snaps. He posted an 80.5 overall grade from Pro Football Focus, ranking tenth among all NFL guards and tenth among all rookies. His 81.5 run-blocking grade led all rookie offensive linemen, while his 68.9 pass-blocking grade ranked third. Over 656 pass-blocking snaps, he allowed only three sacks and thirty-five pressures. Puni was named to the Professional Football Writers of America All-Rookie Team. He is the cousin of Buffalo Bills defensive end A.J. Epenesa.