Rhyl were knocked out of the Uefa Cup in the first qualifying round after a narrow second-leg defeat in Lithuania.
Defender and captain Timmy Edwards conceded a penalty with a clumsy tackle on 19 minutes and Darius Maciulevicius made no mistake from the spot.
Suduva added a second six minutes later when Tomas Mikuckis headed home.
Andy Moran's 80th-minute strike gave Rhyl hope of securing the draw needed to go through on away goals, but Suduva kept out their desperate last efforts.
Lee Hunt, Marcus Orlick and Morgan had gone close earlier before the striker's goal from a goalkeeping mistake set up a frantic final 10 minutes.
The match in Marijampole was soured by a serious injury to Chris Sharp, son of former Scotland and Everton striker Graeme, who was taken immediately to hospital.
Rhyl also had Gareth Wilson sent off in the final minute for a second bookable offence.
FK Suduva: Padimanskas, Grigas, Gledrius Klevinskas, Davit Cigladze, Mikuckis, Maciulevicius, Urbsys, Miklinevicius, Maciulis, Braga, Abramenko.
Subs: Saulius Klevinskas, Adomaitis, Giedrius Slavickas, Besik Cigladze, Vaidas Slavickas, Jasaitis, Cordeiro.
Rhyl: Whitfield, Brewerton, Tim Edwards, Mark Powell, Horan, Graves, Murtagh, Connolly, Wilson, Moran, Hunt.
Subs: Gann, Moody, Sharp, Orlick, Hay, Owen, Rafferty.
Referee: G Mujezinovic (Montenegro).