Golodryga was born to a working-class family of Bessarabian Jews in Căușeni, Moldavian Soviet Socialist Republic (now independent Moldova), the only child of Zhanna and Vitaly Golodryga. In 1980, when she was 18 months old, her family left the Soviet Union as political refugees, each with $75. They settled in Houston, Texas. She has returned to Moldova only once since then, visiting her grandparents and cousins in 1988. That same year, her grandmother joined them in Houston. Her mother is the Chief information officer of Hess Corporation and her father, a mechanical Engineer, was a consultant for DuPont.

Golodryga persisted and was offered a job at CNBC, where she worked for six years as a Producer. She became an on-air correspondent after being asked to fill in one day. In 2004, Golodryga was named one of the top journalists under the age of 30. She was a correspondent for ABC between 2007 and 2010, and was named co-anchor of the weekend edition of Good Morning America in May 2010, following the departure of former co-anchor Kate Snow, who went to work for NBC.

In September 2010, she married Peter R. Orszag, the former Director of the Office of Management and Budget for the Obama Administration and the managing Director and vice chairman of investment banking at Lazard. They have one son and one daughter.

In April 2013, Golodryga was the first person to interview Anzor Tsarnaev, the father of Boston Marathon bombers Tamerlan Tsarnaev and Dzhokhar Tsarnaev, which she called "the most emotional moment" of her career. At first, she assisted her colleagues with the proper pronunciation of the name, but then was tasked with calling Anzor in Chechnya. He had very few details to offer but called her back the next day, trying to find more information about what was happening and whether or not Dzhokhar, who had been taken into custody severely injured, was still alive.

She served as weekend co-anchor of Good Morning America until August 4, 2014, when she left it to join the Business and Finance news department of Yahoo! News. She has guest-hosted on Way Too Early and is a regular contributor to Morning Joe on MSNBC. In 2017 she was a guest co-anchor of the CBS Morning News. She joined CBS on a permanent basis as a correspondent in September 2017 and simultaneously joined CNN as a contributor.

In December 2016, Golodryga confronted Congressman Dana Rohrabacher, a Republican from California, over his defense of Russian Pres. Vladimir Putin. At the time, Rohrabacher was considered a candidate to be appointed Secretary of State by then-President Elect Donald Trump, and was criticizing China for its record on human rights. Golodryga then asked about Putin, who has been consistently praised by Trump. Rohrabacher answered, "Oh, baloney. Where do you come from? How can you say that?" to which Golodryga replied, "I come from the former Soviet Union — that's where I came from. I came here as a political refugee. That's where I came from." Rohrabacher appeared flustered and accused Golodryga of being biased before he compared Putin to Mikhail Gorbachev.