Wales captain Sam Warburton admitted that a draw was "not a good result for anyone" after coming away from Dublin with just a point.
Rhys Priestland appeared to have rounded off an impressive comeback for Wales with a late penalty, but Welsh indiscipline allowed Jonathan Sexton to reply and make it 16-16.
Neither team could find a breakthrough in the final few minutes and Wales will now head home to prepare for a six-day turnaround ahead of facing Scotland next Saturday.
"I was disappointed with first 20 minutes, we talked before the game about having to start well because Ireland always come flying out of the blocks," Warburton told ITV.
"We're pleased that we weathered the storm and managed to fight back.
"A draw is not a good result for anyone. It takes away the possibility of a Grand Slam or Triple Crown, and it means we have got to take another big scalp down the line to win the Championship.
"But we have got three home games now so that's the positive."