Trailing 13-0 at half time due to a try from scrum-half Lewis Jones and eight points from the boot of Wales international Ceri Sweeney, the Irish side did launch a second-half comeback.
Fly-half Miah Nikora kicked three penalties at a wet and windy Galway to earn a bonus point but ultimately they fell short of victory.
Eighth place last season represented Connacht's highest tournament finishing position and they would have envisaged this as a perfect opportunity to kickstart their new campaign against a Blues side missing star players Sam Warburton, Leigh Halfpenny, Bradley Davies, Alex Cuthbert and Jamie Roberts.
However after suffering only their second defeat in their last seven matches at the Sportsground, Elwood was frustrated at his side's inability to keep the ball for a sustained period of time.
He said: "It's disappointing in a home game, dominating a lot of the territory and plenty of possession and to come away with a losing bonus is not really acceptable in a game which we would have targeted.
"It's three points lost but in fairness to Cardiff, they dug it out and took their chances.
"We had plenty of possession but we had 11 turnovers in the opening half and that put us under pressure. We just could not hold on to the ball."
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