This is "The Little Black Bag" by Cyril M. Kornbluth. You can read the full story here.
For the record, it was never published in Omni, but you may have read it in Astounding Science Fiction.
A "physicist" goads his minder into giving him specifications for a
time machine. The faux physicist builds it, and uses it to send a
"doctor" friend's highly automated medical kit into the past (our
present), where it is found by Dr. Full, a physician who has succumbed
to alcoholism and fallen to the bottom level of society. At first
attributing its advanced properties and unfamiliar components to
medical advances made since he last practiced, he uses it to heal a
seriously injured young child. The patient's cynical eighteen-year-old
sister, Angie, discovers the patent application date on one of the
instruments (2450) and is quick to grasp the financial opportunities.
She blackmails Full into taking her on as a partner.
Her first patient sees the sharp instruments and balks at another
treatment. To reassure her, Angie demonstrates their safety by running
a scalpel through her arm without harm. Still unconvinced, the client
requests another test. Back in the future, a technician notes the bag
has been used for murder and deactivates its advanced functions. Angie
runs what has just become an ordinary scalpel across her own throat,
with fatal results.
It might interest you to know that this story has been adapted for TV at least twice
"Night Gallery" The Little Black Bag