The son of a pianist mother and a choreographer father, Yu was born into a family of musicians in Shanghai, China and grew up during the Chinese Cultural Revolution. Yu received his early childhood music education beginning with piano studies from his grandfather, the composer Ding Shande. He continued his piano studies at the Shanghai Conservatory of Music, and also began conducting studies, graduating from the conservatory in 1987. He further studied music in Europe at the Universität der Künste Berlin.
After returning from Europe, Yu was appointed principal conductor of the Central Opera Theatre in Beijing in 1992, serving for three years. He produced operas for the Urban Council of Hong Kong for five years. He was a founder of the Beijing Music Festival (BMF) in 1998, becoming its first artistic director.
In 2000, the Chinese government invited Yu to assume leadership of the China Broadcasting Symphony (also called the China National Symphony). Yu held open auditions, becoming the first Chinese orchestra to recruit all its performers this way. The symphony was renamed the China Philharmonic Orchestra and its reputation quickly improved under Yu's leadership. In 2008, Yu and the China Philharmonic performed for Pope Benedict XVI at the Vatican, as part of increased diplomatic initiative between China and the Vatican. In July 2014, the China Philharmonic was the first Chinese orchestra to perform at the BBC Proms.
In 2003, the Guangzhou Symphony Orchestra appointed Yu its music director. In 2009, the Shanghai Symphony Orchestra appointed Yu its music director. During his tenure, the orchestra has begun its Music in the Summer Air Festival (MISA) in August 2010, constructed a new home for the orchestra (the Shanghai Symphony Hall) in 2014, established the Shanghai Orchestra Academy in 2014, and begun the Shanghai Isaac Stern International Violin Competition in 2016. Yu has shared the position of Artistic Co-Director of the Music In the Summer Air Festival (MISA) with Charles Dutoit since the festival's launch in 2010. In January 2015, Yu was named principal guest conductor of the Hong Kong Philharmonic Orchestra, the first appointment of a mainland Chinese conductor to the position.
Yu and his wife, the violinist Vera Tsu, live in Shanghai. The couple has one daughter.