The deal, worth up to £30 million if targets are met, was announced by Havas – led by its chairman and chief executive, Yannick Bolloré – on Tuesday.
The group has bought 51 per cent of Work Club, with the option to buy the remainder at a later date.
Robertson dismissed speculation that the lengthy negotiations (talks began more than a year ago) were due to concerns about Work Club’s finances.
It was instead attributed to the agency’s obsession with culture. She said: "They wanted to know which people they were going to work with, how we would treat them."
Robertson added: "We have done plenty of acquisitions in the past and we all know the ‘pump and dump’ story. I don’t believe in doing that. In the late 90s, that’s what Havas did.
They bought everything they could and it brought us down." Work Club will be renamed Havas Work Club and report to Robertson.
For global clients, it will report to Havas Worldwide’s global chief executive, Andrew Benett.
Work Club’s strategy partner Paddy Griffith has been promoted to chief executive, while the current chief executive, Martin Brooks, will take on a global development role.
Jon Claydon, the chairman; Andy Sandoz and Ben Mooge, the creative partners; and Lisa De Bonis, the strategy partner, will retain their positions.
Claydon said: "We haven’t achieved what we set out to achieve yet.
This gives us an exciting platform to do that." Work Club will open an office in Shanghai, and possibly one in New York, this year.