The 2002 Rookie of the Year became the second two-time winner on the circuit this year. Fellow young star Kurt Busch also has two victories.
"I'm definitely tired," said Newman, who started from the pole. "My arms hurt, my back hurts, my neck hurts, and I'll be really sore tomorrow and probably Tuesday. But it pays good money."
Newman collected $199,325 for his third career victory as he averaged 106.896 miles per hour. But things did not look good when smoke started coming from his Dodge Intrepid with about 180 laps to go.
"Once I realized it was the power steering, I knew I could overcome that," he said. "This is one of the hardest places to have that problem and overcome it, but we had a fast race car and I wasn't about to pull it in the garage and say that's it."
Newman gave Dodge its first triumph at the spring race at Dover and the manufacturer's first victory here since Richard Petty won in September 1975.
Jeff Gordon finished second and Terry Labonte third, both in Chevrolet Monte Carlos. They continued their solid driving of late as Gordon secured his eighth top-10 finish in his last nine races and Labonte notched his fifth straight top-three.
An angry Tony Stewart was fourth in his Chevrolet. He led for 67 laps before pitting on lap 138, but was penalized one lap by NASCAR officials for stopping inches outside his pit box.
Stewart almost came all the way back from being a lap down and in 32nd place.
"It would be one thing if (the NASCAR official) was out there to tell us to get back, but he wasn't," said Greg Zipadelli, Stewart's crew chief. "That's what I was upset about. If you come over the line, they tell you to stop and go back, and you're OK. I can't see from where I'm at and the tire changer didn't say anything.
"We weren't over the line. We were on the line by an inch. What are you going to do? A rule's a rule."
Stewart vied with Newman for the lead until a caution flag on lap 202.
"From what I was told, he screwed up in the pits and pitted outside of his box," Newman said. "He had to take the one-lap penalty. That's not a piece of tape or a blown tire or something like that. That's his own screw-up. We did what we could to keep him a lap down, knowing he had a fast race car. Thankfully, I think we did, seeing he came all the way back."
Jimmie Johnson, who won both races at Dover as a rookie last year, led 36 laps shortly before he crashed his Chevrolet on lap 277.
While running third, Johnson's car brushed the backstretch wall and rammed into the inside barrier.
"I just lost it off (turn) 2," he said. "Somebody was saying there was some oil down, (but) I'd been pretty loose. I just lost it, chased it and had it saved a couple times but I ran out of race track and brushed the outside wall. When I did that, it turned me back down into the inside wall."
Newman claimed the lead from Gordon on lap 327, but Earnhardt moved ahead 38 laps later after a green-flag pit stop. Newman took over the top spot for the final time with 33 laps to go. Gordon attempted a couple of challenging moves, but Newman held him off.
"I love battles like that, down to the finish," Gordon said. "You have to realize, when you're sitting in my position, I'm in second and I'm looking at inside or outside of him, but I almost lost a position by doing that. I dived in there underneath Ryan and then Stewart got a run on both of us and almost got inside of me. It cost him a position."
Gordon also got into a skirmish with Sterling Marlin. The two came together and Marlin spun off on turn 4 of lap 114. His crash into the inside wall forced him into the garage.
The incident was reminiscent of one two weeks ago in The Winston at Lowe's Motor Speedway.
"We were racing hard and caught the 01 car (Mike Wallace) and got held up by him as usual," Marlin said. "We got clear and the 24 (Gordon) got up under us and turned us around again. That's twice in the last three races. It's kinda getting old. I don't know what's wrong with the 24, but we need to find out."
Gordon claimed the contact was unintentional.
"I definitely got into him, no doubt about it," the four-time NASCAR champion said. "I certainly didn't mean to. I don't know if I caught the apron or just got a big push. We were racing hard for position and the front end just took off on me.
"I tried to stay out of him, but I just couldn't and I clipped him in the left rear. I'm sure he's mad, and I was mad when I got hit by him one time at Richmond."
The race was slowed by caution nine times for 68 laps and twice by light rain early in the race.