Incumbent agency Red, which handled the account for five years, was not involved in the competitive pitch, Get Safe Online said.
Get Safe Online comms director Sarah Martinez told PRWeek the company was "all about offering helpful advice".
She said: "A lot of the safety messaging online can be frightening, which can stop people from acting. We want to change that attitude and help people, not scare them."
Unity's remit will include improving the way safety messaging is communicated by delivering a series of creative campaigns throughout the year. The agency will also support the Get Safe Online press office.
Martinez said there would be a specific campaigning push around October's Get Safe Online Awareness Week.
Unity director Katy Stolliday said the agency would approach the work "through the lens of people’s passion points and bring some real personality to the programme in order to get people to engage with such a rational message".
Unity's Spark Something Good campaign for M&S was named Campaign of the Year - Techniques at the PRWeek UK Awards in October. The agency, which handles PR for clients including Facebook-owned VR business Oculus and Sony Pictures, was also named PRWeek UK Agency of the Year for 2016.