Kavcic broke the serve of 42nd-ranked Haase three times, but the Dutchman eventually came through to line up a match against Chile's Nicolas Jarry.
Brazilian qualifier Thiago Monteiro was on court for 2 hours, 40 minutes before beating Ruben Bemelmans 6-7 (4), 6-3, 7-6 (8).
Monteiro, ranked 124th, will meet U.S. Open finalist Kevin Anderson of South Africa in the second round.
India's Yuki Bhambri ended the hopes of wildcard countryman Arjun Kadhe with a 6-3, 6-4 victory and will play eighth-seeded Pierre-Hugues Herbert of France.
Ilya Ivashka of Belarus earlier beat Sumit Nagal of India 6-3, 6-3, while Laslo Djere of Serbia rallied to beat Marius Copil of Romania 5-7, 6-4, 7-5.