O'Driscoll - the first Irish leader of the Lions since Ciaran Fitzgerald in 1983 - was "the only choice" according to O'Sullivan.
Refering to the captaincy race, the 46-year-old Corkman admitted: "It ended up a one-horse race. But I know Clive (Woodward) talked to Paul O'Connell and I'm sure he spoke with Lawrence Dallaglio too.
"If you were looking for the form guy - the guy who has been at the coal face - then O'Driscoll was the only choice.
"It's going to be a tough test for him but he's equipped to deal with that," added O'Sullivan of the 26-year-old Dubliner.
"If you go through the great captains of recent times - like Keith Wood and Martin Johnson - they've all raised their games at crucial stages and been there when the chips are down."
O'Sullivan - who first handed O'Driscoll the Ireland captaincy at the age of 23 - knows the Leinster centre will rise to the challenge of a six-week 11-game tour.
"I've no doubt he will do well. He has developed as a player and a person over the years and although he's only 26, behind that is a lot of experience.
"He has over 50 caps for Ireland and this will be his second Liosn tour. He uses that experience very well and that will stand him in good stead in New Zealand."