Rabbi Wasserman was born in Biržai (Birz) in present-day Lithuania to Rabbi Naftali Beinish, a shopkeeper and Rakhel Sheina. In 1890, the family moved to Bauska (Boisk) in present-day Latvia, and Wasserman, then 15 years old, studied in the Telshe Yeshiva in Telšiai (Telz) under Rabbi Eliezer Gordon and Rabbi Shimon Shkop. In the summer of 1897, Wasserman met Rabbi Chaim Soloveitchik at a health resort and "became deeply attached to him and his way of learning." He left Telz and traveled to Brest-Litovsk (Brisk) in present-day Belarus, where he learned under Soloveitchik for two years, thereafter considering him his primary rebbe (teacher and mentor).
Wasserman was married in 1899 to Michla, the daughter of Rabbi Meir Atlas, rabbi of Salantai (Salant). Wasserman lived in his father-in-law's house for many years and rejected offers of rabbinical posts (including a prestigious rabbinate in Moscow) being afforded the opportunity to learn Torah at home. He did however decide to teach, and together with Rabbi Yoel Baranchik, he started a mesivta in Amtshilov, Russia in 1903 and earned himself a reputation as an outstanding teacher. Prior to 1907, Wasserman heard that another local rabbi wanted to head the mesivta in Amtshilov and he left to avoid an argument, returning to learn in his father-in-law's house. From 1907 to 1910, he studied in the Kollel Kodshim in the Raduń Yeshiva in Radun' (Radin), headed by The Chofetz Chaim. While at the kollel, Wasserman studied with Rabbi Yosef Shlomo Kahaneman, who would later become the rosh yeshiva of the Ponevezh Yeshiva, for eighteen hours a day.
In 1910, with the encouragement of The Chofetz Chaim, Wasserman was appointed rosh yeshiva of the mesivta in Brest-Litovsk, leading its expansion until it was disbanded in 1914 with the outbreak of World War I; with its closing, Wasserman returned to The Chofetz Chaim in Radun'. When the Eastern Front reached Radun', however, the yeshiva there was closed and Wasserman fled to Russia with The Chofetz Chaim.
In 1914, the yeshiva was exiled to Smilavichy (Smilovichi near Minsk) and Wasserman was appointed its rosh yeshiva one year later when The Chofetz Chaim decided to relocate to Siemiatycze (Semiatitch), and together with Rabbi Yitzchok Hirshowitz (son-in-law of Rabbi Eliezer Gordon from Telz Yeshiva), was asked to keep Torah alive in Smilavichy.
In 1921, after the war, when the Soviet government began permitting Torah scholars to leave Russia, Wasserman moved to Baranovichi, Second Polish Republic (now in Belarus) where he took the lead of Novardok Yeshiva. The yeshiva grew under Wasserman's supervision, and soon had close to 300 students. Copies of the notes taken from Wasserman's Torah lectures were passed around many of the yeshivas in Europe, increasing his influence and fame over most of the Torah world. He was one of the leaders of the Agudath Israel movement and was regarded as the spiritual successor of The Chofetz Chaim.
When there was not enough money to buy food for the yeshiva students, Rabbi Wasserman traveled to America to raise money for the yeshiva. Rabbi Wasserman made an impression on many young Jews that he met while in the United States. Rabbi Wasserman returned to Poland although he knew his life was in danger by returning. This was partly because he did not want to abandon his students, and partly because he took a dim view of American Jewry. In 1939, just before the Nazi invasion, he even forbade his students from accepting visas to the United States to study at Yeshiva University and what is now the Hebrew Theological College due to what he perceived as a spiritually dangerous atmosphere in these two institutions.
When World War II broke out, Rabbi Wasserman fled to Vilnius (Vilna) and in 1941, while on a visit to Kaunas (Kovno), he was arrested by the Nazis with twelve other rabbis and sent to his death.
Rabbi Wasserman was taken and murdered by Lithuanian collaborators on the 12th of Tammuz, 1941, in the Seventh Fort of Kaunas Fortress. Before he was taken he gave this statement: "In Heaven it appears that they deem us to be righteous because our bodies have been chosen to atone for the Jewish people. Therefore, we must repent now, immediately. There is not much time. We must keep in mind that we will be better offerings if we repent. In this way we will save the lives of our brethren overseas. Let no thought enter our minds, God forbid, which is abominable and which renders an offering unfit. We are now fulfilling the greatest mitzvah. With fire she (Jerusalem) was destroyed and with fire she will be rebuilt. The very fire which consumes our bodies will one day rebuild the Jewish people."
Rabbi Wasserman had discouraged emigration to America or Palestine, viewing them as places of spiritual danger. He was particularity critical of the Zionist enterprise in Palestine and claimed "Anti-Semites want to kill the body, but Zionists kill the soul. Better to die than consort with the Zionists."
Rabbi Wasserman had several sons. Rabbi Simcha Wasserman served as Dean of Yeshiva Beth Yehudah in Detroit in the 1940s, founded Yeshiva Ohr Elchonon (later renamed Yeshiva Ohr Elchonon Chabad/West Coast Talmudical Seminary) in Los Angeles, California, in the 1950s, and later founded Yeshiva Ohr Elchonon in Jerusalem. Rabbi Wasserman's other sons were Naftoli and Dovid.
Rabbi Wasserman was famous for his clear, penetrating Talmudic analysis. His popular works, essential material in yeshivas around the world, are unique in their approach. He would often quote his rebbe, Rabbi Chaim Soloveitchik, saying "Producing chiddushim (novel Torah concepts) is not for us. That was only in the power of the Rishonim. Our task is to understand what it says." This approach is evident in his works, which include:Kovetz Heoros
Rabbi Wasserman also published the responsa of the Rashba with annotations in 1932. His talmudic novellae appeared in the rabbinic journal Sha'arei Tzion (1929–1934) and in other publications.