Brooke got on the front foot against Woodward, intimating his 2003 World Cup triumph had contributed to the Lions' downfall in New Zealand in 2005.
"Your ego's Bigga than my head, that's big. U lost 3 nil cause of U," Brooke tweeted.
In his Daily Mail column, Woodward said Ireland's win had the Lions positioned well for next year's New Zealand tour.
"A raft of Ireland players have put themselves very close to the top of the pecking order - Conor Murray, Johnny Sexton, Robbie Henshaw, a rejuvenated Rob Kearney, Jack McGrath, Rory Best, Devin Toner and others. They have set the standards."