Andy McDonald, B.J. Crombeen and David Backes also tallied for the Blues, who opened the 2010-11 season with a 2-1 overtime win against Philadelphia on Saturday. T.J. Oshie registered a pair of assists in the victory.
Jaroslav Halak stopped 13-of-14 shots.
Saku Koivu notched the lone goal for Anaheim, which dropped back-to-back games at Detroit on Friday and Saturday at Nashville. The Ducks will face Vancouver in their home-opener on Wednesday.
"We're getting the puck into the neutral ice, but we're not controlling the puck at all," Ducks head coach Randy Carlyle said about the offensive struggles. "The other team has had the puck more than we have. We're not winning enough one-on-one battles."
St. Louis opened the scoring, striking for two goals six seconds apart early in the first period. After Anaheim failed to clear the puck out of its own zone, David Perron got the puck, skated in on a three-on-two and slid it over to Backes for a redirection tally at 3:53.
Then off the ensuing faceoff, McDonald took a feed from Oshie and fired a wrister that beat the Ducks' netminder to the glove side at 3:59.
"That was big. Kind of gave us an early two-goal lead," D'Agostini said about the quick start. "We didn't stop working from there which was key."
Crombeen's shorthanded tally off another Ducks turnover 5:10 into the second stanza made it 3-0, but Koivu got the visitors on the board 58 seconds later while on the power play when his wrist shot from in close squeezed through Halak.
D'Agostini put the Blues up 4-1 with 4:09 left in the middle frame when he took a rink-wide pass from Alex Pietrangelo and blasted a slap shot over the shoulder of Hiller from the top of the right circle. Hiller was then replaced by McElhinney.
St. Louis had a golden opportunity to blow the game wide open as the Blues were awarded seven minutes in power-play time at 17:00 of the second. Following a major scrum involving several players, Ducks forward Bobby Ryan received a fighting major, instigator penalty and a 10-minute misconduct for his role. However, Anaheim was able to kill off the entire time that carried over into the third period.
D'Agostini's power-play tally on a turnaround wrister from the left circle with 2:42 left in the third accounted for the 5-1 final.
Earlier on Monday, the Ducks signed veteran defenseman Andreas Lilja to a one-year contract...St. Louis forward Cam Janssen did not play after suffering a head injury against the Flyers on Saturday...The Blues finished 1-for-10 on the power play, while Anaheim went 1-for-3...Both teams combined for 143 minutes in penalties...Anaheim and the Blues split four games during last year's season series.