The 21-year-old Lehecka, ranked No. 81, broke four times in the Group C match at Ken Rosewall Arena in Sydney to bounce back from an opening singles loss to Taylor Fritz of the United States on Thursday.
Zverev was competing for the first time since June when he tore three ligaments in his right ankle during his Roland Garros semifinal match against Rafael Nadal.
In other early matches Saturday, Brazil played Norway in Brisbane and Belgium took on Bulgaria at Perth.
Nadal was set to lead Spain against Britain in later Group D matches in Sydney.
Two singles matches – one men’s and one women’s – are played over each of two days, with a mixed doubles to conclude the five-match encounter on the second day.
The inaugural United Cup has teams from 18 countries competing in Brisbane, Perth and Sydney through to Jan. 4. The three city champions and the next-best performing country from the group stage will meet at Ken Rosewall Arena from Jan. 6 to 8 to determine the overall winner.
After matches completed Friday, top-ranked Greece leads Group A with a 4-1 record, Switzerland heads Group B (5-0), the U.S. is first in Group C (4-1), Britain (3-2) tops Group D, Italy is in first place in Group E with a 3-2 record and France (5-0) leads Group F.