Tlaib, the daughter of Palestinian immigrants, said in recent interviews that she favors a one-state solution to the Israeli-Palestinian conflict, as opposed to a two-state solution that would establish a Palestinian state alongside Israel.
Zielinski, a clinical specialist for Allina Health, had hoped to flip a seat that's been held by Democrats since the early 1960s - most recently by U.S. Rep. Keith Ellison, who opted to run for state attorney general this year.
Democrat Rashida Tlaib was elected to represent Michigan's 13th House District on Tuesday, defeating three other candidates, including a write-in campaign by Detroit's City Council president, Brenda Jones.
When the new Congress convenes in January 2019, for the first time it will have at least one Muslim woman among its ranks.
Omar fled Somalia's civil war with her parents at the age of eight and spent four years at a refugee camp in Kenya.
She also recognized Michigan's Rashida Tlaib who is also Muslim for winning.
She won a seat in the state's legislature in 2016, becoming the first Somali-American lawmaker in the country. Her real victory came when she won the Democratic primary.
"I ran because of injustices and because of my boys, who are questioning their (Muslim) identity and whether they belong", Tlaib said in an U.S. television interview in August.
Elsewhere, Democratic Congressman Jared Polis won the governor's race in Colorado, US media projected, making him the first openly gay person to be elected governor in the United States. I will be using my position in Congress so that no country, not one, should be able to get aid from the USA when they still promote that kind of injustice.
Omar campaigned on a similarly progressive platform, which calls for universal healthcare and tuition-free colleges.
Ms Omar did not know English when she went to the USA at the age 12. "It just shows how incredibly wonderful our country can be", she said.