Born in Chernivtsi in the Soviet Union (today in Ukraine), Edelstein was an activist for Hebrew language learning. In 1984 he was arrested by Soviet authorities allegedly for drugs possession. He spent three and a half years in a labour camp in difficult conditions, and was seriously maimed. In 1987 he was released and allowed to immigrate to Israel.
After arriving in Israel he started to participate in political life. Initially a member of the National Religious Party and a vice-president of Zionist Forum, he founded the Yisrael BaAliyah party together with fellow Soviet dissident Nathan Sharansky. He was elected to the Knesset in 1996, and was appointed Minister of Immigrant Absorption in Binyamin Netanyahu's Likud-led government. He was re-elected in 1999, and was appointed Deputy Immigrant Absorption Minister by Ariel Sharon in 2001.
He retained his seat in the 2003 elections, shortly after which Yisrael BaAliyah merged into Likud. Although Edelstein lost his seat in the 2006 elections, in which Likud was reduced to 12 seats (Edelstein was fourteenth on the party's list), he re-entered the Knesset as a replacement for Dan Naveh in February 2007. He retained his seat in the 2009 elections after being placed twelfth on the party's list, and was appointed Minister of Information and Diaspora in the Netanyahu government.