The nearest thing to a confrontation between two pure jets in World War II wasn't air combat- on 19 March 1945, four Arado Ar 234 bombers of KG 76 took off to attack targets around Brussels, Belgium. One of the Arados attacked the Melsbrock airfield and dropped a cluster bomb on a parked Gloster Meteor belonging to the RAF's No. 616 Squadron. Only minor damage resulted.
Source: International Air Power Review, Volume 24. "Air Combat: The Arado Ar 234, the short-lived combat career of the world's first true jet bomber" by Dr. Alfred Price, p116.