Indian Singh, 40, made up five shots in the final round to take overnight leader Molinari of Italy to the play-off.
He then birdied the first hole to seal the trophy and a place at Royal Lytham.
Briton Marc Warren led by three shots with four holes to go, but a double bogey on the 15th paved the way for a collapse that saw him finish level third with Swede Alexander Noren, one stroke behind the top two.
"It was surreal how easy it all was up until that point," Warren told the BBC.
"I was still ahead after the double bogey but I lost concentration on a putt of a couple of feet - and it is one of those things that happens in golf.
"It was all a bit disappointing. It was in my hands with four holes to go, it was up to me but I didn't finish it off."
Noren also faded away after Singh had taken the clubhouse lead.
Briton Matthew Baldwin was tied for fifth with Frenchman Thomas Levet and Dane Soren Kjeldsen after shooting 67, as did Singh.
Luke Donald of Britain and American Phil Mickelson, who had tucked in behind Molinari after the third round, shot 73 and 74 respectively to finish tied for 16th.