Stuttgart, who have six points and are third in Group G, need to beat Unirea Urziceni at home in their final game on December 9 to progress. Romania's Urziceni have eight points, two behind group leaders Sevilla.
Rangers, whose previous Champions League home games this season ended in 4-1 defeats - against Sevilla and Unirea - have two points.
"It is a superb feeling and I cannot get enough of it," Stuttgart manager Markus Babbel told reporters.
"We created good chances and the only slight hitch was maybe we could have scored more."
Rangers striker Kris Boyd passed up a fabulous chance to level six minutes from the interval after Kenny Miller seized on a poor pass from Pavel Pogrebnyak on the right and charged forward to sweep in an inviting cross from the right.
But Boyd planted a seven metre header over the crossbar with only keeper Jens Lehmann to beat.
Rangers manager Walter Smith was left to rue his side's home form in the tournament.
"We have not been good enough when the expectation level has been to win the game," he said.
"We certainly had to struggle at home trying to impose ourselves."
Asked why that was Smith replied: "We are more comfortable away from home when the expectation is not as great to go forward and win the game."
