They began their Pool A campaign well with a 31-19 victory against Malta.
But they then suffered back-to-back defeats to host nation Croatia (21-12) and group winners Switzerland (26-12).
Sweden then went down 40-0 to the Swiss in the quarter-finals.
They bounced back with victory by the same margin against Croatia in the ranking semis, though.
And another blanking came in the fifth/sixth play-off as they crushed Nordic rivals Denmark 43-0.
Switzerland were crowned champions with a 29-0 victory against Monaco.
And Romania took bronze after downing Turkey 24-10.
In the points table, Switzerland sit top with 20, followed by Monaco (18), Romania (16), Turkey (14), Sweden (12) and Denmark (10).
The bottom half reads Hungary (8), Croatia (6), Latvia (4), Norway (3), Malta (2) and Moldova (1).
Sweden head coach Alex Laybourne said: “Finishing fifth and taking two wins on Day Three shows the progress we’re making as a group.
“It is great to see how much we can learn and improve within a tournament.
“And scoring 83 unanswered points in our last two games was an excellent answer to the challenges we faced before.
“There is real disappointment we didn’t deliver more overall, but it’s part of the journey.”
He added: “We’ll take those lessons, keep building and turn our focus towards the second leg in Moldova.”
Sweden head to Chisinau from June 26-28.
*Picture by RugbyEurope
