Byrne, who was not present at the event, earned his prize for his work in the therapy-themed series "In Treatment," while Paquin picked up the statuette for her performance in the vampire drama "True Blood."
"This is awesome! I am so excited," Paquin said as she accepted her award.
"Mad Men" was named best television drama series.
Tom Wilkinson won the supporting actor in a TV series, movie or miniseries honor for his portrayal of Ben Franklin in the historical miniseries "John Adams."
"Well, well, well. Thank you Hollywood Foreign Press Association people. You've made me very happy, indeed," Wilkinson said.
"John Adams" also earned the award for best television movie or miniseries, as well as best actor and actress honors for its leads, Paul Giamatti and Laura Linney, who played John and Abigail Adams.
"This was a hell of a job, this thing," Giamatti said of shooting the celebrated epic.
Laura Dern, Wilkinson's co-star in the made-for-TV film "Recount," picked up the trophy for supporting actress in a TV series, movie or miniseries for her portrayal of Florida Secretary of State Katherine Harris.
"Thank you to the brave and amazing HBO," she said, referring to the pay cable television network that aired "Recount," based on the Florida recount in the 2000 U.S. presidential election.