Miloșovici was born in a family with Serbian roots. Her mother, Ildiko, was a competitive volleyball player, while her father, Tănase, was a national team wrestler. She began gymnastics at the age of 6 and moved to the Deva National Training Centre as she excelled in the sport.
Miloșovici's gymnastics career was nearly derailed when she contracted scarlet fever at the age of 10, and again when the Romanian Revolution in 1989 temporarily closed Deva. However, she continued to train, and by 1990, was representing Romania in such international meets as the American Cup and the World Sports Fair. She competed well at the 1991 Junior European Championships, winning two gold medals.
Miloșovici made her debut as a senior in 1991, winning her first all-around title at the Romanian National Championships. She joined the Romanian team at the 1991 World Championships, where she contributed to the team's bronze medal and won her first individual World Championships title, on the vault, and placed third on the balance beam.
Miloșovici won her second individual world title on uneven bars at the 1992 World Championships. At the 1992 Olympics in Barcelona, Miloșovici won a total of four medals, placing third in the all-around, second with the Romanian team, tying with Hungarian Henrietta Ónodi for the vault gold medal, and winning floor exercise with a perfect score of 10.0. She scored her 10 on the same night that China's Lu Li also scored a 10 (hers being for her performance on the uneven bars). Miloșovici's 10, in the event finals, was only the second awarded in Barcelona, and the last ever awarded at any Olympics.
After the 1992 Olympics, Miloșovici continued to compete, winning the world title on beam in 1993, which completed her quartet of golds on each of the individual apparatuses, and the world silver medal on vault that same year. In 1994 and 1995, she led the Romanian team to two World Championships team titles. Shortly before the 1994 World Championships, she and her teammates staged a strike at Deva to protest nonpayment of prize money owed from the Romanian Gymnastics Federation; in spite of the depleted training time, they still won the team gold medal at Worlds. Milosovici's best all-around finish in World Championship or Olympic competition came in 1994 when she took second to Shannon Miller in a controversial finish, after being considered the early favorite. Milo concluded the 1994 Worlds with a silver on floor and a bronze on vault. At the 1995 Sabae World Championships, she captured the all-around bronze.
Miloșovici led the Romanian team to a bronze at the 1996 Olympics, in spite of a spate of injuries that left the squad depleted of several key gymnasts and forced them to compete with only six athletes instead of seven. At the 1996 Olympics, Milo had the rare opportunity to improve upon her third place all-around finish in Barcelona and only because of her consistency did she take advantage of her competitors' mistakes, and she became the first gymnast since Nadia Comăneci to medal in the all-around at consecutive Olympics. This also marked her fourth all-around medal in World and Olympic competition, a feat since equalled only by Svetlana Khorkina, Aliya Mustafina and Simone Biles.
Miloșovici announced her official retirement in the summer of 1997. After retiring, she returned to Lugoj to coach gymnastics and attended the Sports University in Timisoara.
On 23 October 1999, Milo married her childhood friend, Cosmin Vinatu. According to Prosport, Cosmin was a sergeant major in the police force. The two were childhood friends, but lost touch when Milo moved to Deva. Then, in 1998, Milo was out with her brother and a group of his friends and they ran into Cosmin at a bar. Needless to say, Milo and Cosmin's friendship was renewed. Their wedding took place at the Termal restaurant in Timisoara, with over 350 guests. Milo told Prosport that the couple expected to live in Timisoara after the wedding, though she does own a house in her hometown of Lugoj. They had one daughter, Denisa Florentina, who was born in 2004 and died in 2008. The Godmother was former teammate, Simona Amânar. In January 2013, Lavinia gave birth to a boy, Cosmin Mihai, and in July 2016 Lavinia's second son, Andrew Nicholas, was born.
In 2002, Miloșovici, along with former teammates Corina Ungureanu and Claudia Presăcan, sparked controversy by posing naked in photographs for a Japanese photobook, LCC Gold, and performing gymnastics routines topless for the Japanese DVDs Gold Bird and Euro Angels. A number of photographs from the photobook and DVDs were subsequently published in the Japanese magazine Shukan Gendai, and an edited version of the DVDs entitled 3 Gold Girls was released in Germany in 2004. Because the gymnasts had posed for some of the photographs and footage wearing their official Romanian team leotards, the Romanian Gymnastics Federation banned them from coaching and judging in the country for five years, and the Japanese Junior Gymnastic Club Federation contemplated banning Romanian gymnasts from a major international junior competition.