Tom Mohan's side went behind on the hour mark as Michal Tomič headed home from a corner to give Slovakia the lead.
After conceding, Ireland went on the attack and were awarded the penalty after Jonathan Afolabi was fouled by defender Vladimír Majdan. Connolly cooly slotted past Frederik Valach for a well-deserved equaliser.
Despite late chances for Connolly, Afolabi and Daniel McKenna, Ireland were unable to find a winner.
Speaking after the game, Head Coach Tom Mohan said he was happy to earn a valuable point.
"The boys showed great character in coming back into the game after we conceded. We got our rewards with the penalty and could have won. It's a big point though.
"It was very windy conditions today, and the pitch wasn't the best. The venue was changed at short notice which didn't help preparations.
"The boys showed a lot of composure after conceding. It spurred them on, and we look forward now to the second game."
The team return to action on Saturday afternoon, with a game against Kosovo. That game kicks off at 3pm.
Slovakia: Frederik Valach; Vladimír Majdan, Dominik Špiriak (capt), Denis Petro, Samuel Kozlovský; Martin Bednár, Jozef Špyrka (Branislav Sluka 43); Michal Tomič, Adam Brenkus (Adam Kopas 90+1), Matej Grešák (Marián Chobot 86); Róbert Boženík.
Republic of Ireland: Mark Travers; Daniel McKenna, Canice Carroll, Dara O’Shea (capt), Tyreke Wilson; Conor Coventry (Rowan Roache 67), Thomas O’Connor, Kian Flanagan; Lee O’Connor, Jonathan Afolabi, Aaron Connolly
Referee: Kristo Tohver (EST)