The St. Louis Cardinals' Jake Westbrook, in contrast, has received more than his fair share of run support.
With Dempster and Westbrook both looking to continue their impressive starts, the Cubs open a two-game set against a Cardinals team hoping to snap its longest home losing streak in two seasons Monday night.
After suffering back-to-back losses at Milwaukee, Chicago (14-20) took Sunday's series finale 8-2 - matching its second-highest run total of the year.
That's also as much offense as the Cubs have provided over Dempster's five outings.
While Dempster (0-1, 1.02 ERA) has limited opponents to a .164 average and struck out 36 over 35 1-3 innings, he's still seeking his first win largely because of an NL-worst average of 2.04 runs of support.
"I just try to execute my pitches and control the innings," he said. "The other stuff you can't worry about. You can only control one thing when you're on the mound and that's trying to put the ball where you want to put it. If you do that, more often than not, you'll get good results. The more good results you get, the more wins you'll get."
Dempster allowed six hits and fanned seven over seven innings of one-run ball during Tuesday's 3-1 loss to Atlanta.
"It's hard to believe a guy can have a (1.02 ERA) and not have a win yet in this league," manager Dale Sveum told MLB.com. "It was another dominating performance."
While Westbrook (4-2, 1.76) has also been superb, he's benefited from an average of 8.12 runs of support - the NL's highest mark.
"It lets you settle down and go after guys and be more aggressive, which is what I'm trying to do anyways," Westbrook, who struck out a season-high eight over seven scoreless frames in Tuesday's 6-1 win at Arizona, said of the support. "Hopefully we can continue that trend."
Westbrook looks to help the Cardinals (20-14) regroup from a three-game sweep against Atlanta - their first at home since dropping three against the Los Angeles Dodgers from Aug. 22-24.
"(The Braves) played better baseball than us," Carlos Beltran said after Sunday's 7-4 loss. "They pitched better, they played better defense, they came through when the team needed them the most, and we couldn't do that.
"But you know what? We have to turn the page."
St. Louis, which has allowed 29 runs during its four-game skid at Busch Stadium, is in the midst of its worst stretch at home since dropping five straight Aug. 14-20, 2010.
While the Cardinals had a rough weekend, Sveum knows his club has its hands full - especially when it comes to trying to slow down Beltran and Allen Craig.
Both outfielders went deep Sunday, providing all of St. Louis' offense. Beltran has six long balls in as many games while Craig has five in his last seven.
"Obviously, they're the best team in the league, so we're going to have to shut them down," Sveum added. "They've got an American League offense right now. Beltran is on fire. They've penciled Craig in from the DL ... You really have to keep those guys in the ballpark and it's tough to keep them down."
Chicago and St. Louis have split six matchups this season, with the Cardinals taking two of three at home. The Cubs have dropped nine of 12 in St. Louis.
Since joining Chicago, Dempster has allowed three runs or fewer in all but one of his nine road starts in this series.
Westbrook has a 1.93 ERA while winning his last two outings against the Cubs - including a 10-3 victory April 15.