The Elland Road boss was named on a five-man shortlist after guiding the Whites to an impressive ninth-placed finish this season.
United dazzled and delighted onlookers during their debut top flight campaign for 16 years to earn a top half place.
The top prize, though, has been awarded to Manchester City manager Pep Guardiola following his side's record-breaking title triumph.
City defender Ruben Diaz also claimed the player of the year award for the 2020/21 term.
In 32 league appearances, the 24-year-old Portugal international contributed to 15 clean sheets, made 35 interceptions, 24 tackles, 23 blocks and 135 recoveries.
Last month he was named the Football Writers’ Association’s Footballer of the Year, and he is among the Professional Footballers’ Association’s player of the year nominees.
The nominees Dias beat to take the Premier League award were his City team-mate Kevin De Bruyne, who won it last year, Bruno Fernandes, Jack Grealish, Harry Kane, Mason Mount, Mohamed Salah and Tomas Soucek.
Guardiola, whose side finished 12 points clear of second-placed Manchester United, wins the managerial award for a third time, having also done so on the other two occasions he guided City to the Premier League title, in 2017-18 and 2018-19.
The other nominees alongside Bielsa and Guardiola were Red Devils boss Ole Gunnar Solskjaer, Leicester’s Brendan Rodgers, West Ham’s David Moyes.
Erik Lamela’s rabona effort for Tottenham Hotpsur against Arsenal in March's North London derby was voted the goal of the season.