Nick Punto drove in the other three runs for the A's, who won their fourth straight game after losing five of their previous six.
Kazmir (5-1) did not walk a batter and struck out four as he bounced back from his first loss of the season. He retired 15 in a row at one point.
Norris, who set a career high with his six RBIs, became the first Oakland player to hit a pair of three-run homers on 3-0 counts, and the third in franchise history.
"If you asked me before the game I would have said it feels like I have never picked up a bat before," Norris said. "My pre-game was so terrible I had to leave the cage. I was all out of sync; everything pushed me in different directions when I tried to swing."
Norris, who came to the A's along with Tommy Milone in a trade for Nationals' starter Gio Gonzalez, felt much better once he started seeing pitches. He even took a little extra pride that his first two-homer game was against his former team.
"Most definitely," Norris said, "And against the guy I was traded for. It was definitely a little weird playing against the guys you played with coming up."
Brandon Moss added three hits, while Punto, Jed Lowrie and Craig Gentry each had two as the A's collected 13 hits, nine against Gonzalez, who was making his first start against his former team.
A's manager Bob Melvin implied that might have been part of the thinking in giving Norris the green light on a 3-0 pitch.
"We don't do it a whole lot, especially against starters," Melvin said. "I think more than anything, probably the first time, I think Gio's been around here and knows that we don't do that a whole lot. Derek's been seeing the ball really well so I trusted him to take it if it was a ball. You know you're going to get a fastball, and make sure you get it in a good spot."
Gonzalez (3-3) retired the first two batters of the game before walking Josh Donaldson. Yoenis Cespedes followed with a single and Norris stepped in to slam his third home run of the season, a ball that traveled halfway up the left-field bleachers.
"Would you wait for that pitch? Especially with two men on base," Gonzalez said. "I would have done the same thing."
Norris, who did not receive a shipment of pink bats, still honored his mother, Jacque, with his best game as a professional.
"She has always been there for me," Norris said. "She watches every game I play. I'm not superstitious; give me a stick of wood and I'll go up and hit."
The A's added another run in the first when Alberto Callaspo followed Norris with a single and eventually scored on Punto's first hit.
Gentry opened the second with a single and stole second base. He advanced to third on Lowrie's fly to right. After Donaldson was retired, Cespedes walked.
Gonzalez again went to a 3-0 count on Norris, who hit the next pitch nearly to the same spot as his first home run.
"Just couldn't find the strike zone," said Gonzalez, who received a warm welcome from the A's fans. "That was my mistake, just trying to figure it out. Mistake pitches, left them up in the zone and good hitters made contact."
Norris also faced a 3-0 count against Gonzalez in the fifth, but walked on a pitch in the dirt.
Punto added a two-run double in the seventh.
Nearly lost in the offensive show was Kazmir's gem of a performance.
"It's something where the first pitch of the game I felt like I was right on point," Kazmir said. "Location-wise, everything was right there. I was able to attack the strike zone in on righties a lot and I think that helped me."