Hengelbrock was born on June 9, 1958, in Wilhelmshaven, in what was then West Germany. He studied the violin with Rainer Kussmaul, a member of the Berlin Philharmonic. Early in his career, he was active as a violinist in the cities of Würzburg and Freiburg. Becoming interested in conducting, he worked as an assistant to conductor Antal Doráti and to composers Mauricio Kagel and Witold Lutoslawski, developing an interest in contemporary music that he would maintain throughout his career. Attracted by the growing historical performance movement, he co-founded and directed the Freiburger Barockorchester in Germany (with which he made his recording debut in 1990, leading a performance of C.P.E. Bach's Hamburg Symphonies on the Deutsche Harmonia Mundi label), and he also conducted and appeared as a soloist with the pioneering group Concentus Musicus Wien, working directly under its director, Nikolaus Harnoncourt. In 1988, he became the director of the Amsterdam Bach Soloists, remaining in that post until 1991.
In that year, Hengelbrock founded the Balthasar Neumann Choir, adding to it a Balthasar Neumann Ensemble instrumental group in 1995; he remains the director of both. Named for a famous Baroque-era architect, these groups focus on historical performance but have a repertory that includes contemporary works. Hengelbrock served as the artistic director of the Kammerphilharmonie Bremen from 1995 to 1998 and was music director of the Volksoper Wien from 2000 to 2003; he also guest-conducted opera in various European houses. In 2001, he founded the Feldkirch Festival in Austria, serving as its director until 2006. Hengelbrock became principal conductor of the NDR Symphony Orchestra in Hamburg, overseeing its move into the Elbphilharmonie concert hall and its renaming as the NDR Elbphilharmonie Orchestra. Hengelbrock's recordings with that group focused mostly on mainstream symphonic repertory. He stepped down from that position at the end of the 2017-2018 season but has remained active in the recording studio, issuing albums on Sony Classical as well as Deutsche Harmonia Mundi. In 2022, he and the Balthasar Neumann Orchestra backed soprano Julie Fuchs on the Mozart recital album Amadè on Sony Classical. ~ James Manheim, Rovi