After Craig Dargo had put the visitors ahead with almost their only effort of the first half, substitute Steven Thompson pulled Rangers level shortly after half-time.
The home side pinned Inverness back for long spells, and this intensified when Graham Bayne was sent off for a second booking seven minutes from the end.
However, Mark Brown in the visitors' goal defied them to the end with a brilliant finger-tip save in the final minute.
Peter Lovenkrand was first to threaten for Rangers as he weaved past three defenders but shot narrowly wide.
Darren Dods tackled Barry Ferguson superbly at the edge of the box before Dado Prso created space and then watched aghast as no-one read his cut-back.
Inverness took the lead with a simple goal. Brown's long clearance was missed by Marvin Andrews, and Dargo hit a first time shot into the top corner from 25 yards.
The closest Rangers came to an equaliser was when Fernando Ricksen headed Olivier Bernard's cross against the bar.
Rangers drew level soon after the break. Alan Hutton's cross spun high into the air off Dods and Brown, and Thompson was first to react, shooting home from sixyards.
It looked inevitable that the home team would go on to win, but as chance after chance was squandered, their hopes flagged.
Prso glanced a header inches wide and Thompson and Andrews were denied at the near post.
In the final minute Brown produced that last-minute save which must put manager Alex McLeish's future in doubt.