She later had supporting parts in Clean and Sober (1988) and When Harry Met Sally... (1989), before starring in the major studio films Parenthood (1989) and Arachnophobia (1990).

[3] Kozak starred in movies such as The House on Sorority Row (1983), Parenthood (1989), Side Out (1990), Arachnophobia (1990), The Taking of Beverly Hills (1991), Necessary Roughness (1991), All I Want for Christmas (1991), The Favor (1994), Magic in the Water (1995), Unforgivable (1996), and the soap operas Texas (from November 1981 to December 1982 as Brette Wheeler), Guiding Light (from 1983 to 1985 and a one-day voice-over in February 1990, as both Annabelle Sims Reardon and Annabelle's deceased mother, in 1983 flashbacks, Annie Sims) and Santa Barbara (from 1985 to 1986 and again in 1989, as former nun, Mary Duvall McCormick).

In Santa Barbara, her character died in an accident where a giant neon letter "C" toppled onto her during an argument atop the Capwell hotel.

The show received such a huge number of letters that eventually they admitted their mistake and asked Kozak to come back.

She played the role of Bess Allison, a mother who dies while searching for her missing baby at the time when RMS Titanic sinks.

Also in 1996, Kozak portrayed a battered wife in a made-for-TV movie, Unforgivable, co-starring with John Ritter, and received praise for her "strong" performance.