The temporary power company confirmed that longstanding executive Angus Cockburn, who has run the company since the departure of Rupert Soames for outsourcing company Serco, did not apply for the permanent position and is leaving the company.
Aggreko said that Mr Weston will be based both in Glasgow and in London.
Mr Weston, who oversaw Scottish Gas and British Gas in his role at Centrica, said: "Aggreko is a fascinating business and I'm very excited to be taking on this role.
"I have watched and admired Aggreko's tremendous growth over recent years."
Mr Cockburn is to leave Aggreko later this year, having worked there for 14 years.
He is tipped to join Mr Soames at Serco.
His permanent replacement as finance director has been announced as Carole Cran, who has been doing the job on an interim basis.
She will join the board of the company with immediate effect. Ms Cran has worked at Aggreko for a decade, including as group financial controller.
Ken Hanna, chairman of Aggreko, said: "Chris brings senior experience of the energy industry and proven leadership skills in a large international business.
"He has consistently succeeded in driving performance and growth in his career and we believe he has the skills and vision to continue Aggreko's growth and lead the group to the next level.
"In addition I would like to thank Angus Cockburn so much for the immense contribution he has made to Aggreko over the last 14 years.
"We can understand his desire to seek a fresh challenge and wish him every success in his future career. He has an extremely able successor in Carole Cran who has many years of experience with the group."
Mr Weston is currently managing director, international downstream at Centrica, running the group's largest division.
This makes him responsible for Scottish Gas and British Gas in the UK and Direct Energy in the USA, with revenues of £22 billion and profits of £1.3bn.
He joined Centrica in 2001 after a career in the telecoms industry, when the group bought One.Tel.
Before that, Mr Weston served in the Royal Artillery before earning a science degree as well as an MBA and PhD from Imperial College London.
He is reported to have been interviewed for the top job at Centrica, to replace incumbent Sam Laidlaw, but was not appointed to the post.
It could be 2015 before he joins Aggreko because he is subject to a 12 month notice period.
Mr Laidlaw said: "Chris has made an important contribution to Centrica over 11 years, leading a number of our businesses, in the UK and North America.
"We will be sorry to see him leave, but recognise that Aggreko offers a great opportunity for him."
Iain Conn, a BP executive, is understood to have been offered the post of Centrica chief executive.
Aggreko has a power projects arm that has relatively long term contracts with developing countries who have not developed the capacity to cater to rising demand.
Its local business handles short-term contracts catering to events such as the football World Cup and the Olympic Games.
In the first quarter of 2014, Aggreko reported a 5 per cent rise in underlying revenue supported by strong global oil and gas activity. But it has been hindered by a drop in demand from the Australian mining industry.