Inverness had taken the lead in the 49th minute when Andrew Shinnie finished off a move he had started himself.
But Hearts fought back strongly and Michael Ngoo netted a debut goal 24 hours after signing on loan from Liverpool.
Robinson was sent off for a two-footed challenge in a full-blooded encounter that also saw nine players booked, but the Gorgie side held on for penalties.
Inverness substitute Philip Roberts missed the final spot-kick to spark Hearts celebrations at Easter Road.
- Sports & Recreation
- Andrew Shinnie