The Chiefs looked to have earned themselves a 6-6 draw in what was their first ever game in the competition before Jonathan Sexton kicked over in the final few minutes in Dublin to deny them a point.
"We put everything we had into the game, we've come up a little bit short but there's a lot there for us to be positive about and a lot to build on," Hayes told Sky Sports News.
"We've a lot of belief in ourselves in what we're about as a team. We've grown a bit as a team together over the last three or four years and we're developing all the time so today is just another step forward."
Exeter will hold a Heineken Cup match at Sandy Park for the first time on Saturday when they host Clermont Auvergne.