Brandon Inge belted a two-run homer off Zambrano in the seventh to go ahead, 3-2, but Micah Hoffpauir's two-run homer with two outs in the eighth gave Chicago a 4-3 lead. In the ninth, Gregg walked Don Kelly and Raburn, a .225 hitter, launched a slider into the left-field seats for the game-winner.
"I could've hit that ball out of the park," Gregg said. "It slipped out of my hand. I was hoping he'd pop it up. He popped it up over the fence. It [stinks]."
Zambrano was in line for the win.
"Things happen in baseball," Zambrano said. "Not even K-Rod or Mariano Rivera in save opportunities are good sometimes. They struggle, too. Gregg is a human, too. Sometimes he'll give it up, and sometimes he'll be there for us. Today he gave it up. Tomorrow he'll be there to save the game for us."
The Cubs had plenty of chances. They stranded six and went 0-for-8 with runners in scoring position.
"We've got one-run leads," Lou Piniella said. "On the road, it's not the easiest thing in the world."
Having Zambrano in the lineup to hit may have helped. He struck out four and gave up three runs on five hits over seven innings. Before the game, the switch-hitting right-hander pretended to be upset at not being in the lineup.
"[Designated hitter] -- no respect," said Zambrano, who walked into Piniella's office with his bat, trying to convince the Cubs manager to ignore American League rules. Not to worry -- Zambrano didn't take any swings at the manager. He did miss his at-bats.
"A lot," Zambrano said. "Not only to be able to hit, but to be able to get going and keep my routine. That's why I took [batting practice] today. Coming to the American League is boring for me. You feel like you're the DH. You go out there for an inning and come back to the dugout and do nothing."
The Cubs, hitting a sluggish .236 on the road, loaded the bases in the first and took a 1-0 lead when Derrek Lee hit into a forceout at second. The Tigers answered in their half when Curtis Granderson doubled to lead off and scored on Placido Polanco's triple.
Milton Bradley walked to lead off the Chicago sixth and reached third on Lee's double, which extended his hitting streak to 20 games, the 22nd such streak in franchise history. One out later, Geovany Soto flew out to center to score Bradley.
Zambrano walked Kelly to start the Tigers' seventh, and Inge followed with his 17th home run to put Detroit ahead, 3-2.
Joel Zumaya took over for Detroit starter Edwin Jackson and retired the first two batters on 100-mph fastballs. Lee then singled, and Zumaya changed speeds and threw an 85-mph changeup to Hoffpauir, who connected for his sixth home run and the lead.
"I got lucky, I guess," Hoffpauir said. "I keep going back to the first two [at-bats]. We shouldn't have been in that situation at the end of the game. I put a lot on myself. Geo picked up that one the second time, but I could've easily had four RBIs."
Hoffpauir struck out for the second out in the first to strand two and struck out again in the sixth when there were two on and none out. It's not just him.
"We're professional hitters," Hoffpauir said. "We're supposed to be able to go out there and do the job."
That's what Piniella is thinking, too. He cut short a question after the game.
"I don't want to talk about missed opportunities, I really don't," Piniella said.
Zambrano's next start will be Sunday against the White Sox, again in an AL ballpark. He does have a clever idea to help him get the win.
"I was thinking about eliminating the seventh inning and going straight to the eighth," Zambrano said. "The last two seventh innings against the White Sox and Detroit have been no good for me. I think I'm going to jump that inning."
Carrie Muskat is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.