It is being performed at Curve theatre in Leicester, the city where Townsend lived and where the book is set.
The writer worked on the musical before her death last April and her family have given their blessing to the production going ahead.
There will be more than 30 performances over the next four weeks.
The author's family were at Curve for the opening night.
Her husband, Colin Broadway, said his wife had been "thrilled" to be working on the musical during 2012-14.
The role of Adrian is being shared by Lewis Andrews, Sebastian Croft, Joel Fossard-Jones and Toby Murray.
The roles of Pandora, Nigel and Barry are each being shared by three actors.
This has meant everything has needed rehearsing four times.
Executive producer Chris Stafford said: "For us this is something that is the culmination of three and a half years' worth of work, but also we really hope it to be a fitting tribute to Sue, that will really introduce Adrian Mole hopefully to a new generation of audiences."

Gwilym Joseph MBE served the club for nearly 50 years and was given the honour earlier this year.
A statement from the club said it was in recognition of his "tireless work and commitment" and said his energy and enthusiasm was "boundless".
Mr Joseph, who died early on Friday, was also the club lead on building the Liberty Stadium and the training grounds at Landore and Fairwood.
Swansea chairman Huw Jenkins said: "It's a very sad day for the club.
"We will always be extremely grateful for his commitment, knowledge and dedication to this football club.
"On a personal note, I have to say that ever since I have been chairman he was always a trusted ally and supporter of my work, something I will always be eternally grateful for.''
The club stalwart, awarded his MBE in the 2014 Honours List for services to football and the economy in Swansea, was born in Morriston in 1927.
Between the ages of 14 and 16 he set up the only youth club in the area, which opened at Pentrepoeth School in 1941.
Having established two football teams within the youth club - Morriston Athletic and Morriston United - he continued to support them and various charities through fundraising dances.
He also formed the club's Executive Fund Raising Committee in 1974 which has so far raised more than £500,000.
He leaves his wife, Jan.
A musical version of The Secret Diary of Adrian Mole Aged 13ยพ has opened as a tribute to the book's author Sue Townsend.
[NEXT_CONCEPT]
The honorary president of Swansea City FC has died aged 90.