The 21-year-old American had lost both previous meetings with the 6-foot-10 Karlovic, who won the only U.S. men's tournament played on clay in 2007 at a different Houston venue.
In this one Sock broke the Croatian's huge serve twice in the first set and wound up spending less than an hour on the court.
German-born Dustin Brown also easily moved on, beating American Rhyne Williams 6-1, 6-2. Williams had reached the tournament's semifinals last year. The 29-year-old Brown, who began his pro career playing out of Jamaica, is ranked 101st.