The poll, from Forsa poll for broadcasters RTL and n-tv, suggested they had moved ahead of the ruling Christan Democrats (CDU) and coalition partner the Social Democrats with 27 percent of voter intentions.
It came just a week after the party unexpectedly came second in the European elections, with 20.5 percent of the vote.
This poll puts the Greens nine points up on polls from a week ago, with the Christian Democrats and their Bavarian regional partner CSU down two points at 26 percent.
Chancellor Angela Merkel's CDU were the largest party in the EU polls, but on 28.9 percent — their worst-ever score in a national election.
The SPD, an unhappy partner in the national government, and two years out from the next scheduled general elections, was down five points down in the latest poll at just 12 percent.
In the EU vote the SPD obtained 15.8 percent.
The opinion poll also showed the extreme right AfD moving to within one point of the SPD on 11 points, which is what they scored in the European elections.
State broadcaster ARD television has in recent days reported that climate change has become a priority for 46 percent of voters — up 26 percentage points since the 2014 European elections.
Recent polls also suggest a generational split, with the Greens leading the CDU among those aged 18-44, and those over 60 favouring Merkel.
Young voters in particular — energised by the Fridays for Future school strikes, anti-coal protests and a passionate campaign from leading YouTube stars — abandoned the mainstream CDU/CSU and SPD parties in droves.
In the wake of her party's comparatively poor showing in the European elections, Merkel on Wednesday promised a new climate strategy by September.