The result means Ireland will now face Wales in the final group game to decide the winner after the Welsh drew with the Czech Republic.
Ireland were gifted the best chance of the game with just one minute on the clock as Megan Connolly closed the keeper down but after winning the ball, her shot was blocked.
Ireland continued to create chances as the impressive Harriet Scott, in just her second appearance for the national side, burst down the left to find Aine O'Gorman with a low cross, but the UCD Waves striker saw her shot well saved.
A minute after the break, the tireless Katie McCabe worked a neat position just outside the area after a weighted-flick from Stephanie Roche, who claimed her 50th cap, but her powerful effort was well saved.
Hungary could have won the game with just under 30 minutes to go as they were awarded a penalty after Aine O'Gorman fouled Gabriella Toth in the area but Fanni Vago's penalty went well wide of Hourihan's right-hand post.
After the final whistle, Head Coach Colin Bell was delighted with the character and the effort shown from the group.
Bell said: "I'm really pleased with the group. It was a really intensive match against a really strong side in Hungary who have played together for some time.
"They stuck to the game-plan and even when they made mistakes, the effort levels to recover was fantastic. We should have been 1-0 up at half-time if we took our chances.
"We had a little bit of luck in the second half with the penalty miss but I think we deserved as Hungary didn't create another chance in the entire 90 minutes.
"We're trying to lay the foundations. We want to stay compact and get teams on the break. When we start training together and the team gets fitter, we can then start moving the ball better and become a much better side."
Republic of Ireland: Hourihan; Perry, Caldwell, Quinn, Scott; McLaughlin (O’Riordan 73), Duggan; McCabe (McCarthy 90), Connolly (Kiernan 84), O’Gorman; Roche.