Aggreko's finance director Angus Cockburn will be interim chief executive while the Glasgow-headquartered group launches an internal and external search for a replacement.
Mr Soames, the grandson of wartime prime minister Winston Churchill, said: "I have loved every minute of my time with Aggreko.
"But there comes a time when all CEOs need to move on and now, with a new five year strategy in place and an exceptionally strong executive team running the business, that time has come."
Mr Soames, who said last year that "I don't feel like I'm on the way to the glue factory any time soon," now has the challenge of turning around a company that has seen its share price fall 40% since July after, along with G4S, Serco was shown to have charged for tagging criminals who were not being monitored.
The Government then banned it from receiving new contracts.
Mr Soames leaves a company whose shares have risen by a multiple of more than 11 during his tenure, even after lacklustre performance over the past 17 months, thanks to demand from emerging states and contracts for prestigious events such as the 2012 London Olympic Games and Glastonbury festival.
This has allowed him to cash in share awards worth many millions of pounds and he retains a stake in Aggreko valued at more than £5 million.
After the announcement of his appointment, shares in Aggreko closed down 70p or 4.3% at 1560p. Serco enjoyed a 12.1% bounce with its stock finishing up 49.7p at 460.5p.
Mr Soames will receive no exit payment but Serco is giving him shares worth upwards of £1.3m to buy him out of awards he will relinquish on leaving Aggreko.
Mr Soames, who faced controversy over a £7.1m pay packet in 2012, will be paid £850,000 at Serco, up from a basic salary of about £675,000 at Aggreko.
He will also get £255,000 in pension contributions, and be eligible for a bonus of up to 150% of his salary plus shares worth up to 200% of salary.
Aggreko chairman Ken Hanna said: "When your CEO has done 11 successful years, you must expect that he is coming towards the end of his time."
But he added; "I was surprised at the timing. I thought we would have a bit longer from Rupert."
He continued: "He has decided that this allows him to take on one more big executive challenge."
Aggreko gave extra duties to three executive directors 15 months ago as a means of "testing them out" as the company considered succession planning.
Before joining Aggreko in 2000, Mr Cockburn was managing director of Pringle of Scotland,
Mr Hanna insisted that Aggreko remains strong despite a faltering share price since 2012 as emerging market growth slowed.
Serco chairman Alastair Lyons said: "Rupert is a passionate believer in public services, highly respected by customers, governments, employees and investors."
Mr Lyons said that Serco consulted the Government, which provides 25% of its revenues, about Mr Soames's appointment but the final decision was made by the board.
Mr Soames said: "I am aware that the company has experienced significant recent difficulties but the work that Serco does is important to the lives of millions of people and I believe that we can find a way through to a bright future."