Kraft said the decision by the pro-business Free Democrats (FDP) to go into the opposition meant caretaker conservative premier Jürgen Rüttgers could no longer govern the state.
She will now attempt to be elected NRW's premier on July 13 or 14.
"Red-Green will now ensure governable conditions," said Kraft, referring to the two parties' colours.
The decision ends nearly two months of political limbo in the state after an inconclusive election left no clear coalition option.
“Now we can get to work,” said Sylvia Löhrmann, the state parliamentary leader of the Greens.
The SPD and Greens fell just one seat short of a majority in the May 9 poll, however, they were unable to forge a three-way alliance with either the socialist Left party or the FDP. Kraft also rejected forming a so-called grand coalition with the conservative Christian Democrats.
The Social Democrat was at first reluctant to form a minority government, but the national SPD leadership exerted pressure on Kraft not to leave Rüttgers in power because it would leave Chancellor Angela Merkel's centre-right coalition at the federal level with a majority in the upper house of parliament.
With Kraft governing NRW, Merkel's government will have greater difficulty pushing through legislation that requires the approval of the Bundesrat, which represents Germany's 16 federal states.