Williams and Lydiate have been passed fit for the clash, with Lee Byrne and Ryan Jones making way, while Jones comes into the second row for Bradley Davies and Jonathan Davies is preferred in the centres to Scott Williams.
Head coach Warren Gatland has also made two positional changes, with George North switching from the left to the right wing and Leigh Halfpenny starting at full-back.
On the bench, James Hook gets the nod ahead of Stephen Jones after returning to full fitness.
"This was a tough selection for us, probably one of the longest selection meetings we have ever had, where we went through every possible permutation available to us," said Gatland.
"It's a great position to be in when you are struggling to find a place for players in the squad and the starting line-up, instead of having to think who can make up the numbers, which has sometimes been the case.
"This is Rugby World Cup quarter-final time and we had some tough decisions to make with the likes of Dan, Shane and James returning from injury and the side having done so well in their recent absence.
"But this is the side we have gone with and, as a 22, we believe it is the side which gives us the best chance of winning this game against an Ireland team which not only topped their group but beat the Wallabies on the way."
Wales: 15 Leigh Halfpenny, 14 George North, 13 Jonathan Davies, 12 Jamie Roberts, 11 Shane Williams, 10 Rhys Priestland, 9 Mike Phillips, 8 Toby Faletau, 7 Sam Warburton (captain), 6 Dan Lydiate, 5 Alun Wyn Jones, 4 Luke Charteris, 3 Adam Jones, 2 Huw Bennett, 1 Gethin Jenkins. Replacements: 16 Lloyd Burns, 17 Paul James, 18 Bradley Davies, 19 Ryan Jones, 20 Lloyd Williams, 21 James Hook, 22 Scott Williams
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