Cabrera, the 2007 U.S. Open and 2009 Masters champion, almost earned a second green jacket at Augusta earlier this year, losing in a playoff to Adam Scott.
Crane, a Westlake resident, had his best finish of the season at the Shell Houston Open, where he tied for fourth.
Harrington, a three-time major champion (2007-08 British, '08 PGA), will make his second appearance at the Nelson after finishing tied for 15th in 2012.
The field will not be final until Friday at 5 p.m. The HP Byron Nelson Championship will be played May 13-19 at the TPC Four Seasons Resort Las Colinas in Irving.