Niemeyer, ranked 487th in the world, earned a standing ovation from a healthy court one crowd with the victory.
Niemeyer's serve was exceptional, scoring seven aces and hitting on 66 per cent of his first serves.
Up 3-1 in the second set, Niemeyer was facing break point but fought back to deuce with an ace and won the next two points easily with strong first serves to keep Kunitsyn on his heels.
Earlier Monday, 65th ranked Russian Mikhail Youzhny knocked off 14th seed Marin Cilic of Croatia 4-6, 7-6 (4), 6-1.
The towering, six-foot-six Cilic with the booming serve did not take advantage of his main weapon, hitting only 50 per cent of his first serves.
It is the second straight tournament of the hardcourt season where Cilic did not win a single match, as he was also knocked off by 153rd ranked Somdev Devvarman of India in Washington, D.C. last week.
In other singles action early Monday, Germany's Rainer Schuettler easily handled France's Florent Serra 6-4, 6-0. Schuettler, ranked 91st in the world, will face seventh-seeded Jo-Wilfred Tsonga of France in the second round.
Russian Igor Andreev advanced to round two when countryman Dmitry Tursonov retired due to a left ankle injury with the score tied 5-5 in the first set.
It is a big day for the Canadian entries, as Frank Dancevic will also play Monday while Peter Polansky faces Bruno Agostinelli in an all-Canadian match.
