There were also differing fortunes for the Northern Irish teams as Dungannon Swifts beat FK Suduva 1-0 as Glentoran were comprehensively defeated 5-0 at home by AIK Solna. In the Republic, St Patrick's Athletic and Drogheda United were both held to draws in their first legs.
Haka went ahead after a quarter of an hour through a Toni Lautinen counter, but Andy Moran after the Finns failed to clear a free-kick.
Lee Hunt put Rhyl ahead just before half-time and it was his left-wing cross that set up Craig Garside to score on his debut.
The return leg is on August 2nd.
It was a night to forget, meanwhile, for Carmarthen, who were trounced by a very strong Brann Bergen side full of international quality, including former Scotland man Robbie Winters.
Winters helped himself to a hat-trick, while Thorstein Helstad scored two and Kristjan Sigurdsson, Jan Solli and Armann Bjornsson all got on the scoresheet.
Neil Smothers was dismissed for the Welsh team for two bookings.
DRAWS FOR THE IRISH
St Pats ground out a goalless home draw with Danish side Odense while Drogheda conceded a late equaliser against part-timers SP Libertas from San Marino to draw 1-1.
St Pats will be happier with their result as Odense had the better chances, Bechara Oliveira going close on several occasions.
Anthony Murphy fired wide from an early corner for St Pats while Ryan Guy was denied by Anders Christensen.
Oliveira was the real threat though, with Bolanos also threatening. The second leg is in two weeks time.
Drogheda, meanwhile, will be going back to Ireland scratching their heads as to how they did not win, dominating the match against Libertas but somehow failing to add to Eamon Zayed's goal just before the break.
Aidan O'Keeffe, Sami Ristila, Shane Robinson, Stuart Byrne and Richie Baker all went close but goalkeeper Michele Ceccoli was in fine form and Federico Nanni punished them after a counter by Federico Santini on 77 minutes.
MIXED FORTUNES IN NORTH
A Mark McAllister goal was enough for Dungannon to beat Suduva at Windsor Park.
The strike on 18 minutes was provided by Shea Campbell. Both sides could have scored further goals.
The tie now hangs in the balance, with the second leg in Lithuania to come.
Glentoran were hammered 5-0 by Swedish club AIK Solna at home. Wilton Figueiredo opened the scoring midway through the first half and there was no way back for the Glens.
Striker Figueiredo made it two on the hour mark as Gary Smyth hit the bar for the home side.
However further strikes from Lucas Valdemarin, Khari Stephenson and Dulee Johnson left them almost certainly without a chance of progressing after the second leg in Sweden.