Inter Turku are the second Finnish team to begin their UECL campaign with an away draw, after Ilves' match away at Differdange in Luxembourg ended goalless.
But they had hopes of leaving the Bosnian capital with much more, after going ahead within eight minutes as Loic Essomba teed up Alie Conteh to finish.
In response, Sarajevo enjoyed the bulk of possession and conjured up more attempts than their visitors, but a lack of cutting edge meant only once in 90 minutes did they force Eetu Huuhtanen into a save.
However, they only needed one more opportunity, as in the 94th minute, substitute Bartol Barisic found a way through to equalise and ensure the sides travel to Turku for the return leg on 16th July with everything to play for.