The appointment follows a pitch that included the incumbent All Response Media, which had held the account for six years, Total Media and Media Campaign Services.
The win means Mindshare has moved swiftly to replace the Everest windows and home improvement account, which it lost to fellow WPP agency MediaCom in August.
Mindshare will work on a campaign for Anglian's new EcoHome product.
Martin Troughton, the Anglian marketing director, said: "We felt Mindshare offered us a unique blend of brute force buying power with a deft touch of creativity. We look forward to a long and fruitful partnership."
Nick Ashley, a partner at Mindshare, said: "As the country's leading company in this sector, we look forward to helping it cement and extend its market leadership over the next few years."
The Point will continue to handle creative for Anglian. The company's relationship with its roster of digital agencies, Bloom Media and Latitude, is also unaffected by the review.