Wales almost repeated that feat at the start of the second half as forward Ffion Morgan finally found some space, but she curled wide.
The hosts were improved after the interval and substitute Carrie Jones forced Brosnan to save from a long-range effort after Hayes’ last-gasp block denied a Rhiannon Roberts header.
With Fishlock dropping into a more central role Wales looked more controlled and Fishlock’s 25-yard freekick was saved by Brosnan as the hosts pushed for a winner.
However, it was Ireland who came closest to a second goal as Hayes shot on the turn as Wales failed to clear a set-piece and Clark produced a brilliant save to turn the ball over the crossbar.
Neither side was willing to risk over-committing to score a winner due to the risk of conceding ahead of the second leg, but it was Wales who came closest as substitute Kayleigh Barton’s injury-time cross just evaded Jones.
Wales manager Rhian Wilkinson on BBC Radio Wales:
"To go into this second leg all square is not a bad situation. Obviously we would have liked to win. It’s half-time.
"My team battled well. I wasn't as pleased with our style of play. We were too direct. Ireland are very good at what they do.
"They don't mess about with the ball, and I think we played into their hands so I expect more from the team in Dublin.
"This team likes to do things the difficult way. The team thrives in adversity so I look forward to seeing an incredible performance in Dublin."
Republic of Ireland manager Eileen Gleeson told BBC Sport Wales:
"It was definitely a battle. It was what I was expecting. It was evenly contested.
"I thought we started better then they came into it for the next 15 minutes or so. It kind of ebbed and flowed.
"There was a mistake for the goal but I'm proud of the girls for coming back and equalising.
"We always said it wouldn't be won or lost tonight. I'd prefer a win but, with a draw, we're in the game and we take it home. It's positive.
“We’d like to think we have the advantage now… going to our home, we love playing at the Aviva Stadium."