"It's pretty crazy," Dempster said. "I think as a reliever, when you do something like that, it seems more realistic because you're going out there one inning at a time. But to go out there start after start and not give up any runs, it's pretty humbling. I'm just trying to get outs and win ballgames, I'm not expecting that."
Dempster's career-best fifth consecutive win marked the Cubs' 11th victory in their last 15 games. The Cubs have won five straight games at Wrigley Field and are 21-20 at home. It's the first time they've been above .500 at home since April 2011.
Manager Dale Sveum said before Saturday's game he wanted to keep Dempster between 85 and 95 pitches. He stuck to his word, pulling Dempster after 89 pitches.
The Cubs' starter allowed four hits, while walking three batters -- including D-backs starter Joe Saunders while leading 3-0 in the fifth inning. Dempster allowed three total walks in his three June starts, before being sidelined with right lat tightness.
"It looked like he was kind of losing it there when he walked the pitcher," Sveum said. "Then, all of a sudden, he came back with probably his best stuff all day. We got him right to the 90-pitch mark in six innings this time, so it all worked out as much as you draw it out."
Dempster said it's easy to lose command coming out of the All-Star break. But just as he did in his return from injury in five scoreless innings on July 8 against the Mets, Dempster rebounded after a brief hiatus and found ways to keep his opponents from scoring.
"It looked like he was laboring a little bit and I thought we were going to have a chance to maybe take advantage of some of the pitches up in the zone," said D-backs third baseman Geoff Blum. "But he just never caves in."
The Cubs gave Dempster a three-run cushion by taking a 3-0 lead in the fourth inning on an RBI double by Darwin Barney and an RBI single by Luis Valbuena, who also scored in the third inning to put the Cubs on the board and has hit safely in eight of his last 10 games at Wrigley Field.
Dempster struck out five batters and relied on his defense when he got into jams. The Cubs have turned at least one double play in each of their last nine games and added three on Saturday.
"That's the thing about Demp," Barney said. "He's been around a long time. He's smart. He knows how he wants to get guys out. He's not out there trying to strike everybody out. He knows how to pitch to contact really well, and that's why he's so good."
After Dempster exited, reliever James Russell allowed a solo home run to Chris Young in the seventh. The Cubs led, 4-0, at the time, and are 27-15 when scoring at least four runs.
Shawn Camp tossed a scoreless eighth inning to extend his scoreless-innings streak to 13 1/3. After Carlos Marmol put two runners on in the ninth inning, right fielder Reed Johnson ran into the wall but still caught a fly ball and Anthony Rizzo started the Cubs' final double play of the day to end the game.
"Rizzo kind of secures up that corner a little bit, making some tough plays look pretty easy," Barney said. "We're really happy with the way we're playing defense."
"Defensively, knock on wood, it's about as good as it can get right now," Sveum said.
Barney, who increased his errorless streak at second base to 74 games, said he looks forward to every day Dempster pitches because he knows he'll get a lot of action. He hopes he'll get to play behind Dempster for the rest of the year.
"He was one of those guys that really helped me out in a lot of situations coming up to the big leagues last year and the year before," Barney said. "He's a leader. He's just a great guy to be around, a competitor. The day he throws, it's absolutely different."
Dempster has tried not to get wrapped up in trade speculation, but he does look back and think about how lucky he's been to play in Chicago for nine years.
"I have been really fortunate that this city has been tremendous to me and my family, and I am so thankful for every opportunity," said Dempster. "And I'm trying to give my best effort when I am out there, and have truly enjoyed every minute of it."
Now on pace to finish with an ERA below 3.00 for the second time in his 15-year career, Dempster doesn't want to think about playing somewhere else.
"I just really enjoy pitching for the Cubs right now," he said.