Rovers, who qualified for the group stages of the Europa League last season, were the better team on the night but struggled to create too many clear cut chances.
They did get closer in latter stages of the match and in the final 10 minutes star striker Gary Twigg miss controlled from six-yards out and Ronan Finn smashed a shot against the crossbar.
It was a similar story for Welsh side The New Saints as they also drew 0-0 at home to Swedish side Helsingborgs.
Again, they were the better team, with new signing Chris Jones having their best chances.
Jones hit the post on one occasion but the Swedish champions were glad to get away with the draw.
The second legs take place next week.
Shamrock Rovers (Ireland) 0-0 Ekranas Panevezys (Lithuania)
Valletta FC (Malta) 1-4 Partizan Belgrade (Serbia)
Skenderbeu (Albania) 1-0 Debrecen (Hungary)
The New Saints (Wales) 0-0 Helsingborg (Sweden)
Zilina (Slovakia) 1-0 Hapoel Ironi Kiryat Shmona (Israel)
Slovan Liberec (Czech Republic) 1-0 Shakhtyor IK (Kazakhstan)
F91 Dudelange (Luxembourg) 1-0 Salzburg (Austria)
HJK Helsinki (Finland) 7-0 KR (Iceland)