He was backed by McLaren team boss Martin Whitmarsh, whose drivers Jenson Button and Lewis Hamilton both had tire troubles on the Bahrain International Circuit.
Team boss Ross Brawn said he regretted not putting Button on a different strategy that might have enabled him to get further up the field early in the race.
Klien, who will take part in a race weekend for the first time since the 2007 British Grand Prix when he replaced Jenson Button in second practice at Silverstone while at Honda, is eager to impress his new boss.