The victim was stabbed inside a sports bar on Sneden Avenue in the Annadale section around 3:45 a.m.
He was being treated at Staten Island University Hospital for a non-life threatening injury.
The suspect fled the scene in a 2003 Maroon Lexus GS 300.
Police believe they caught up with him on Narrows Road North in the borough's Arrochar section, where he may be barricaded.
They blocked off streets around the house as they tried to apprehend the suspect.
The motive for the stabbing is not known.