The agency won the account following a competitive pitch process.
It has been tasked with running a number of consumer campaigns for the rest of this year and into 2015.
Charlotte Speedy, director of comms at Dogs Trust, said: "We want to continue to raise awareness about dog welfare issues in the UK and to promote responsible dog ownership.
"When appointing a PR agency to manage this campaign, we needed to be sure that the ideas were fresh and exciting, reflecting the character of Dogs Trust, as well as the seriousness of our work."
Suzi Williams, senior consultant at Surname & Surname, said: "Working with one of the most well-known charities in the UK, highlighting the issues facing dogs and dog owners, is something that we feel passionately about."