The Icelandic side will enter Tuesday’s contest with a slender advantage after winning 1-0 in the Republic of Ireland last week.
Breidablik had to negotiate two preliminary round ties against Tre Penne and Buducnost to secure their place in the first round of qualifying.
The current Icelandic champions enjoyed a successful trip to the Tallaght Stadium last week, claiming a narrow 1-0 victory thanks to a first-half strike Damir Muminovic.
They followed that result with another 1-0 win in Friday’s league fixture with Fram, representing their third consecutive victory in all competitions.
Breidablik will now turn their attention to Tuesday’s crucial encounter, where they will be looking to win a two-legged Champions League qualifying tie for the first time in their history.
Their only previous appearance in a two-legged Champions League encounter took place in the second round of qualifying in 2011-12 against Rosenborg, which resulted in a 5-2 aggregate defeat.
Shamrock Rovers, meanwhile, are enduring a poor spell of form, having failed to win any of their last three competitive matches.
With no game at the weekend, the League of Ireland leaders have had nearly a week to prepare for the return leg where they know only a win will do.
They have tended to struggle on their travels in UEFA competition, having won just two of their last 14 away European ties.
The visitors may be able to draw inspiration from their most recent trip to Iceland in June 2017 when they beat Stjarnan 1-0 in Europa League qualifying.
If the Hoops manage to turn around the tie on Tuesday then they will advance to the second round of Champions League qualifying for a second straight season.
Breidablik Champions League form:
Breidablik form (all competitions):
Shamrock Rovers Champions League form:
Shamrock Rovers form (all competitions):
We say: Breidablik 1-1 Shamrock Rovers (Breidablik to win 2-1 on aggregate)
After coming up short in the first leg, we expect Shamrock Rovers to produce a battling performance in Iceland, but with just two wins from their last 14 European away games, we think that they will be held to a draw, which would secure Breidablik a place in the second qualifying round.