Players said they spent a lot of time thinking of Hancock, but were able to focus on the game once it started.
"It's tough, but when we're out there on the field, you need to be competitive," Albert Pujols said. "Obviously, that's how Josh was. He was a competitive pitcher. Even if we were winning 20-0 or losing 20-0, he went out there and took the ball. He went out there and pitched great and enjoyed it."
Prince Fielder doubled and homered in his first two at-bats as Milwaukee scored four runs in the first three innings off Cardinals starter Kip Wells (1-5).
"Obviously, it's something that everybody is aware of and it's something we're all dealing with, having a day to get our heads in order," Wells said of his teammate's death. "It's tough, but when you take the field you are able to focus on what you need to do to get people out."
Wells pitched six innings, giving up seven runs on six hits and three walks while striking out three to lose his fourth consecutive start.
"There's not a whole lot of reasons for why things didn't go our way other than they made better pitches than we did and Soup pitched a good game," Wells said. "It was what it was."
La Russa said the game was typical of how the Cardinals have played much of the season.
"The whole first month has been a little fuzzy," he said. "We're not sharp in our focus, whether it's our hitting or pitching. We need to be sharper in our focus. We're playing a little fuzzy.
"That's why our record's not good," he said.
Suppan (4-2), signed as a free agent by Milwaukee during the offseason, gave up one run on eight hits and one walk while striking out five. He also earned the victory in Milwaukee's 3-2 victory over the Cardinals at Busch Stadium on April 14.
"We can be tougher outs," La Russa said. "Not to take credit away from the pitcher because he deserves credit, but we can be tougher outs than we are being."
St. Louis scored its only run in the fourth on Chris Duncan's RBI single, scoring David Eckstein from second.
The Cardinals honored Hancock by hanging No. 32 jerseys in the bullpen and dugout and wearing a No. 32 patch on their left sleeve. In addition, a moment of silence was held before the game.
Three Cardinals -- Pujols, Jim Edmonds and Jason Isringhausen -- were on the team when Darryl Kile died on June 22, 2002, in Chicago. St. Louis lost five of seven games after Kile's death, but rebounded and won the National League Central that year.
"The best way to honor Josh and DK is to go out there and play hard," Pujols said. "They were both pretty competitive."
Pujols singled in the first inning to extend his hitting streak to nine games.
Right-hander Dennis Dove, recalled from Triple-A Memphis to take Hancock's spot on the roster, pitched a scoreless inning in his Major League debut.