Woods, who will be aiming for a fifth Masters title since 1997 and an 11th win in Grand Slam events, tees of at 10:23 (14:23 GMT) on Thursday and has a late start at 13:30 (17:30 GMT) on Friday.
The 25-year-old Molinari gained entry to the tournament by winning the US Amateur Title in 2005, the first European to do so since 1911.
It will be a Masters debut for the slightly built, Turin-born golfer whose brother Francesco is a professional on the European Tour.
Woods said that he had met Molinari for the first time on Tuesday morning as the starting times were being announced but he had been impressed with him in the few holes he saw him play at the US Amateur.
Allenby will be playing in his seventh Masters, but has yet to record a top 25 finish. He did, late last year become the first player to win the Australian Open, Australian PGA and Australian Masters titles in consecutive weeks.
The two met in the second round of the World Golf Championships Match-Play event at Carlsbad, California, in February when Woods came out on top 1-up after a hard-fought tussle.
The top grouping of the day looks set to be 2004 Masters winner Phil Mickelson alongside South Africa's Ernie Els and Shingo Katayama of Japan.
Mickelson captured his first green jacket two years ago after an epic Sunday struggle over the last nine holes with Els, who has yet to win the tournament despite, coming close on several occasions.
Opening the way at 08:00 will be on-form Australian Rod Pampling in the company of Americans Lucas Glover and Olin Browne.