Wales became Guinness World Records champions after France lost 27-23 to Scotland.
The Blues needed an extra point win and a winning margin of at least 21 points after a reorganized clash with the Scots on Friday night to clinch the title, but they failed.
The result in Paris was good news for the Welsh, who are claiming championship glory for the second time in three years.
Wayne Pivak’s men had to wait six more days to know their fate following the grief of Saturday’s 32-30 final defeat at Stade de France, where they were denied a grade slam.
Pivak told BBC Sport: “What a tournament it was, from a Welsh point of view we are in heaven. We are very happy to have won the championship.
“It’s a shame we couldn’t do this as a group last week. We think we tried hard enough to win the game, but we didn’t.
“It doesn’t detract from the fact that we did a lot of work. We came out on top and won the tournament.”
Wales’ victory marked a significant change for head coach Pivak, who has been skeptical of his future since finishing fifth in six countries last year.
In 2020, Wales have won just three of their last 10 matches and are yet to qualify for the tournament.
But after beating 14 men from Ireland and Scotland, a 40-24 win over defending champions England and a regular win over Italy were enough.
“We are very happy with the way things are going in our camp and we built very well during the competition,” said New Zealand’s Warren Gottland, who took charge after the 2019 World Cup.
“We have played a great rugby in this tournament and it is an exciting time for everyone.”
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