On Monday, the Rays ace was selected as the Bank of America American League Player of the Week for the period ending May 21.
After being notified of his accomplishment, Kazmir was content, but again recognized that there is a lot more work to do.
"That is great right now, but I can't think about it too much and sit on it," Kazmir said. "[I] just have to keep going out there and working hard."
In two starts last week, Kazmir went 2-0 with a 0.60 ERA and struck out an AL-high 19 batters as he improved to 7-2 on the season.
The 22-year-old became the youngest pitcher in 25 years to have seven wins this early in a season.
On Sunday, he struck out a career-high 11 batters and shut out the Marlins over eight innings for his fifth straight victory.
Against Chicago, Kazmir fanned eight and gave up just five hits in a 10-7 win. During April, his ERA was 3.72, and after his four wins in May, he has improved it to 2.39.
Kazmir has given up one earned run or fewer in his last five starts, dating back to April 30 against Boston. He has seven of the Rays' 20 wins this season.
The Rays won back-to-back series last week for the first time all season, beating the White Sox and Marlins.
He joins the AL's two most recent winners, Seattle's Joel Pineiro and Toronto's Roy Halladay, to mark the first time that three pitchers have earned AL Player of the Week in three consecutive weeks since 1993.
"[It's] a great feeling to get notoriety like that," Kazmir said. "[I] just have to keep it going."
In September 1993, three pitchers -- New York's Jim Abbott, Minnesota's Kevin Tapani and Kasas City's Kevin Appier -- won three weeks in a row.
Other candidates for the award were Boston's Mike Lowell, hitting .400 with three homers; and Chicago's Tadahito Iguchi, hitting .333 with three homers.
Freddy Garcia went 2-0 with a 1.84 ERA for the White Sox, and Kevin Millwood went 2-0 with a 1.29 ERA for the Rangers.