Coach Mauricio Pochettino's side produced a display brimming with energy and creativity and more than merited their victory, which was sealed when Malik Tillman's late free-kick doubled their lead.
However, it took another moment from Balogun to set them on their way against a pragmatic opponent, who, until the 24-year-old was dismissed just after the hour mark, exhibited little attacking intent.
Having reacted quickest to steer Tillman's deflected pass past Bosnia goalkeeper Nikola Vasilj, the Monaco forward's evening was abruptly cut short when he became entangled with Tarik Muharemovic and his studs raked down the defender's calf towards his ankle.
That moment briefly threatened to spoil the mood for the majority of a partisan home crowd inside the San Francisco Bay Area Stadium, but Tillman's curling effort all but ended Bosnia's hopes of a recovery.
The United States will now face Belgium in Seattle on Monday, 6 July (01:00 BST on Tuesday) for a place in the quarter-finals.
-BBC
Latest News
U.S.-Iran deal unlikely to restore oil and gas flows
Sri Lanka tourist arrivals decline in June
Government targets 7–8% economic growth via technology and manufacturing sectors
Bids called for the Central Expressway Kurunegala–Dambulla stretch
Special Cabinet Subcommittee begins El Niño preparedness planning
LB Finance secures 35 million dollars in long-term funding
Construction sector enters expansion territory
Kane double rescues England against DR Congo
USA advance past Bosnia to last 16
Pathadumbara NS wins U-16 girls’ 4x800m at Hiru Relay