The company has asked the agency to develop a long-term digital strategy to build and maintain customer trust after it received a record £12 million fine for mis-selling.
Engine will oversee all digital and social media duties, including user-experience, creative and mobile activity.
Engine won the brief after a pitch that also involved SapientNitro and Dare. DigitasLBi previously held the account but did not repitch.
Anthony Ainsworth, the marketing director at E.ON, said: "Engine stood out for us throughout the pitch process. It demonstrated a genuine desire to help improve our customers’ experience.
"We’ve had board-level commitment and engagement throughout, and have an extremely talented team of experts in place to help us ensure we meet and exceed our customers’ expectations."
In May, E.ON was fined and forced to apologise to customers for "large scale mis-selling", following an investigation by the industry watchdog Ofgem.
DigitasLBi and Dare declined to comment.
SapientNitro was unavailable for comment.