PHOENIX -- Ted Lilly rebounded nicely after a couple of rough recent starts.

Unfortunately, it wasn't enough to get a Cubs win.

Arizona's Doug Davis outdueled the Chicago southpaw as the D-backs took the second game of this three-game series, 3-1, on Saturday night in front of 46,173 at Chase Field.

Lilly (13-7) had given up 11 runs over his last two starts, but held the D-backs to two runs on just four hits over six innings. He struck out eight and didn't walk a batter.

"He pitched well," said manager Lou Piniella. "Two runs in six innings, I'll take that every time."

"I threw the ball better than I have the last couple of starts," said Lilly. "I was locating the ball [inside] better and my curveball was more consistent than it has been. It's something to take into the next start, but it's still a loss today."

The Cubs' offense could never really get anything going off of Davis.

"He's a crafty left-hander," Piniella said.

Davis (11-11) allowed just one run in seven innings of work, also fanning eight.

"It's kind of unfortunate I didn't keep up with Davis," said Lilly. "He was throwing the ball well. I'm pretty disappointed."

Lilly had faced the minimum number of batters over the first 3 2/3 innings, but Eric Byrnes singled with two out on the fourth, and Conor Jackson followed with his 10th homer of the season to give the D-backs the lead.

"I made some mistakes, and that was one of the mistakes I didn't get away with," said Lilly. "It was a 2-2 fastball, and I was a little too concerned with the runner on first ... I ended up leaving it out over the plate."

The Cubs cut the lead in half on an Aramis Ramirez two-out RBI single in the sixth, but Justin Upton answered for the D-backs with a two-out RBI double off Kerry Wood in the seventh.

The Cubs had some opportunities in this one but couldn't come up with the big hit. They left two men on in the sixth and eighth innings, as Craig Monroe (0-for-4 in his Cubs starting debut) made the final out in each frame. They also left a runner on second in the seventh.

"We had a couple of chances, and that was it," Piniella said "We just couldn't get anything else in."

"Games like that, a lot of times it's one swing of the bat," D-backs manager Bob Melvin said. "[Jackson's] hit was big, Upton's hit was big, and usually in games like that the team who gets the hits with runners in scoring position wins."

Piniella tried his best to get something going, such as calling for a squeeze bunt with a runner on third and one out in the third inning. Lilly was at the plate, but he fouled it off and later struck out as the Cubs eventually left the man on base.

"He got a good pitch but fouled it back," Piniella said.

"It was a great opportunity," Lilly said. "It was a four-seam fastball right down the middle. It's not like it was a real tough pitch to bunt."

Jose Valverde pitched a 1-2-3 ninth for his Major League-leading 39th save.

Although the Brewers lost again to keep the Cubs lead at a season-high 1 1/2 games, Piniella didn't want to hear it.

"We care about what we do," Piniella said. "As long as we continue to play well, we don't have to worry about anything, and if you don't play well it doesn't matter what anybody else does."