It was Sveum's first victory since he was hired to manage the Cubs in November.
"It's better than being 0-3," Sveum said. "You score eight, nine, 10, 11 runs all the time, you're bound to win one."
Soriano led off the Cubs' four-run second inning with a drive that hit halfway up the scoreboard in left field. He led off the third with a double to the wall in center and added a two-run homer in a four-run fifth.
"I'm a little surprised," Soriano said. "It's only the second game and I'm starting to feel good with my swing and my timing. Normal spring, it takes me like 20, 25 at-bats before I feel good."
Guillermo Moscoso (1-0) gave up four runs and five hits in two innings for Colorado.
Cubs right-hander Matt Garza (1-0), who is being considered for the opening-day start, allowed two runs and three hits over two innings in his spring debut.
"He gave up a couple of runs, but at least the ball was coming out of his hand (well), and he was keeping the ball down," said Sveum, who added that Garza didn't use his slider as he worked on his changeup. "I think he probably could have got out of those jams if he'd used his slider. He looked sharp."
The Cubs' top outfield prospect, Brett Jackson, hit a three-run homer in the second inning and added a walk. He has made an impressive early showing, including a pair of home runs in two intrasquad games over the weekend.
"I don't think you're ever going to have a kid like that not play," he said. "He hasn't played a full year of Triple-A. He's either going to make the team and play every day, or he's going to be in Triple-A."
Moscoso, who's competing for the Rockies' fifth-starter job, walked two in addition to five hits allowed in his two innings, on a day no Rockies pitcher escaped unscathed.
"Overall we didn't pitch particularly well today," Rockies manager Jim Tracy said. "That goes without saying. But on a lot of the other things we did today, I thought we had a very good day."
That included left fielder Tyler Colvin, who was acquired from the Cubs in a four-player trade in December, going 2-for-3 with a triple and run-scoring groundout.
Rockies center fielder Dexter Fowler added a home run off the scoreboard in three at-bats.