As a child, he studied piano and violin; he was a very good pianist, making his debut in that capacity with Beethoven's Piano Concerto No. 3 in C minor. Unfortunately, a hand injury ended his piano carreer.
After World War II, Skrowaczewski became music director of the Wrocław Philharmonic, then the Katowice Philharmonic, the Krakow Philharmonic and the Warsaw National Orchestra. He studied composition with Nadia Boulanger in Paris. In 1956 he won the Santa Cecilia Competition for Conductors.
At the invitation of George Szell, Skrowaczewski conducted the Cleveland Orchestra. In 1960 he was appointed music director of the Minnesota Orchestra, a position he held until 1979 when he was awarded conductor laureate. In 1981 the American Composers' Forum commissioned the Clarinet Concerto, which Skrowaczewski wrote for Minnesota Orchestra principal clarinettis Joe Longo, who premiered it in 1981.
From 1984 to 1991 he was principal conductor of the Hallé Orchestra.
Skrowaczewski's Passacaglia Immaginaria, completed in 1995, was nominated for the Pulitzer Prize in 1997. Commissioned by the Minnesota Orchestral Association to honor the memory of Ken and Judy Dayton, it was premiered at Orchestra Hall, Minneapolis in 1996.
The Chamber Concerto was commissioned by the St. Paul Chamber Orchestra in memory of Leopold Sipe, their first music director. The Concerto for Orchestra got a Pulitzer nod in 1999.
He received the Commander Order with White Star, the highest order conferred by the Polish government, as well as the Gold Medal of the Mahler-Bruckner Society, the 1976 Kennedy Center Friedheim Award.