Leo, 18, reached the second round with a 6-3, 6-7 (2/7), 6-0 win over 17-year-old Marko Topo of Serbia.
Borg fired six aces and 34 winners in his win out on Court Nine.
Bjorn Borg won the boys tournament at Wimbledon in 1972 before winning five successive men’s titles from 1976-1980.
Only one father and son have previously won the boys title at Wimbledon — India’s Ramanathan Krishnan in 1954 triumphed as did his son Ramesh Krishnan in 1979.
Leo Borg also played Wimbledon juniors in 2019 but was defeated in the first round.
“It’s special to be here at Wimbledon, my dad played here so many times and won many titles," said Borg.
“We talked before the match, he was just saying ‘good luck’, he lets my coach do the coaching stuff. So he just said ‘it’s always special to play at Wimbledon, so enjoy it’."
Bjorn Borg, now 65, won 11 majors in his career — five at Wimbledon and six French Open titles — and also reached the number one ranking.
But his son insists he does not try to put pressure on himself by thinking about his father’s record.
“I just do my own stuff and go down my own path," he said.
“My goal is to be a professional tennis player, to play at high levels. It’s tough to get there but I’m doing my best."