Lambert is back in the technical area following a seven-month exile after accepting the job with the Staffordshire strugglers, in a move which has left many Stoke supporters underwhelmed by the appointment.
The 48-year-old has signed a two-and-a-half year contract and will be assured of a significant bonus if he can maintain the club's place in the top-flight, with his first game against Huddersfield this weekend.
Lambert's arrival ends a nine-day search for Mark Hughes's successor, with moves for Gary Rowett, Quique Sanchez Flores and Martin O'Neill all ending in frustration.
Rowett was a leading contender but opted to sign a new deal at Derby while Flores, the Espanyol manager, performed a surprise u-turn despite the Stoke board flying out to meet him in Barcelona.
O'Neill, the Republic of Ireland manager, was also interviewed twice by chairman Peter Coates and his son John, the vice-chairman, plus chief executive Tony Scholes, but neither party was ever truly convinced.
It left Stoke's with no alternative but to consider other candidates but Lambert is understood to have impressed so much in his interview that he quickly emerged as the No. 1 choice.
During talks he outlined his ambitions for the season and Stoke's strengths and weaknesses, with his appointment confirmed on Monday morning. He will be assured of funds for the January transfer window.
Peter Coates said: “Paul greatly impressed us with his knowledge of our squad and had a clear plan of how he would improve our results. He’s a man who backed himself as a player, none more so than when he turned down contract offers in Scotland to go on trial in the German Bundesliga with Borussia Dortmund, and it’s obvious he adopts the same approach as a manager.”
There is an acceptance within the Stoke boardroom that the appointment will not be wholly popular, but Lambert has proven Premier League experience and, crucially, was never relegated during a turbulent spell at Aston Villa.
His reputation was arguably at its highest nearly seven years ago, when he achieved back-to-back promotions at Norwich City. He departed for Villa in 2012 and endured a difficult two-and-a-half years under former chairman Randy Lerner, operating on a restricted budget.
Yet he never suffered relegation in that time before his dismissal in February 2015.
Since leaving Villa he has managed Blackburn Rovers and Wolves, while he was interviewed last month by Championship clubs Hull City and Sheffield Wednesday.
January 2018 transfer window
John Coates said: “Paul has been successful in management at clubs with a strong and stable background and with local ownership – the kind of foundation we are able to give to our managers. We were determined to appoint someone with Premier League experience or an extensive knowledge of English football. We look forward to working with Paul and supporting him to help achieve our goals.”
Lambert was in the stands at Old Trafford for Stoke's game against Manchester United on Monday night. He will be officially presented at the bet365 Stadium on Tuesday afternoon.