Alvarez hit long solo shots in the second and fourth off Shelby Miller (0-1) and broke open a close game with a rare stolen base he converted into a run that broke open a close game.
Travis Ishikawa added his first major league home run in nearly two years for Pittsburgh. Gerrit Cole (1-0) gave up two runs in seven innings in his first start since losing the deciding Game 5 of the NL division series to St. Louis last October.
Miller struggled in his 2014 debut. He tied a career high by giving up five earned runs in 5 1-3 innings, including the three home runs. Matt Carpenter hit a two-run homer for the Cardinals.
Six months removed from a taut NLDS that pushed the eventual NL champions to the brink of elimination, the Pirates sent an early message they have no plans of going away.
Cole is considered the ace-in-waiting after a stellar rookie season in which he was arguably Pittsburgh's best pitcher over the final six weeks. Eager to take the next step in his development, the 23-year-old kept the Cardinals in check by mixing a fastball that consistently reaches the upper 90s with improved offspeed stuff.
He also showcased some of his maturity, working out of a bases loaded situation in the fourth by getting Jhonny Peralta to fly out. His only mistake came in the fifth when Carpenter drilled a shot into the right field stands to pull the Cardinals with 3-2.
Pittsburgh's offense - which mustered all of 20 hits in 35 innings of a season-opening series against the Chicago Cubs - made sure it didn't stay close for long after pounding out 16 hits in eight innings against the NL Central favorites.
Alvarez began the year in a 1 for 15 slump but ended that with a home run to the gap in left-center for his first home run of the season. He added his second two innings later with a long drive to right-center that traveled an estimated 448 feet.
Yet Alvarez's most important contribution may have been a trio of sprints in the sixth. The burly third baseman chugged down the first base line to beat out a potential double-play ball to keep a rally going then promptly stole the fifth base of his career. A moment later Alvarez was deftly avoiding Yadier Molina's tag at home plate to score on a single by Russell Martin that made it 4-2.
St. Louis manager Mike Matheny came out to talk to the umpires but did not ask for a challenge - replays showed Alvarez was safe - and the Cardinals unraveled. Martin scored when Molina hit him in the back with a throw during a rundown between third and home and pinch-hitter Gaby Sanchez followed with an RBI single to make it 6-2 before things got out of hand late.
Miller, who went 15-9 in his first full season in the majors, fell to 0-5 with a 5.92 ERA against Pittsburgh after allowing six hits with three walks and two strikeouts.
NOTES: Ishikawa's solo home run in the fourth was his first since May 15, 2012 while playing for the New York Yankees. ... The start of the game was delayed 1:05 by rain, the third straight game a Cardinals' game was delayed by weather. ... Injured Cardinals 2B Mark Ellis is starting baseball activities but it remains uncertain when he'll be ready to return from left knee tendinitis. Ellis went on the 15-day DL on March 27 and is eligible to return on April 11. ... Pirates RHP Stolmy Pimentel, who got his first major league win in Wednesday's 16-inning victory over Chicago, was held out for a second straight day and won't be available until Sunday. ... The series continues on Saturday. RHP Joe Kelly will make his season debut for the Cardinals against Pittsburgh's Francisco Liriano (0-0, 0.00).