It is the first political poll conducted entirely since he became leader.
The poll also shows a marked dive in the popularity of John Key, to the lowest level since he became Prime Minister.
Labour has gained 6.8 points to 37.7 per cent of decided voters which would give it 48 seats.
With the Greens' 14 seats and the Mana Party's one seat, it would be able to form a government of 63 seats in a Parliament of 124 seats.
Polling began two days after Mr Cunliffe won the leadership contest against Grant Robertson and Shane Jones.
From the moment David Shearer resigned on August 22 to the declaration of the new leader on September 15, the contest was conducted in a blaze of publicity.
The rise of Labour appears to be mainly at the expense of National, which is down 5.1 points since the June poll to 43.7 per cent support.
National would not have enough support to form a government with present support partners Act holding the same number of seats as they do now.
And even if its other support partner, the Maori Party, kept its present three electorate seats, National's 56 seats plus five for its support partners would still not be enough to form a government.
Mr Key's support as preferred Prime Minister has fallen 9.4 points to 55.8 per cent.
Since becoming Prime Minister he has been in the 60s and sometimes 70s. He has never been down to the 50s.
Mr Cunliffe scored a respectable 16.8 per cent in his first appearance as Opposition leader in the preferred Prime Minister stakes.
His predecessor's rating in June was 12.4 per cent. A total of 20.4 per cent of those polled said they were more likely to vote Labour next election with Mr Cunliffe as leader than before, 14.4 per cent said less likely, and 63 per cent said it would make no difference to their vote.
New Zealand First gained 4.4 per cent support in the poll, putting it below the 5 per cent threshold. This means that unless it won an electorate, it would have no seats.
Leader Winston Peters gained 6.2 per cent support as preferred prime minister.
National may see the poll result as the result of saturation attention on Labour but it also highlights one of National's biggest electoral problems - the lack of support partners.
The seat calculation figures assume - as have other polls - that electorate seats held by Act, United Future, Mana and the Maori Party are retained. But that is not nearly as certain as it has been for previous elections.
National has faced fewer controversies in recent months than it did at the start of the year, but the Opposition has been accusing it of "crony capitalism" over issues such as the $30 million payment to Pacific Aluminium Smelter, possible help for Chorus through internet pricing and the SkyCity convention centre deal.
National has passed the GCSB legislation, faced uproar over limits to snapper catches, and has made progress on its next part-sale of state-owned energy company Meridian.
The drop in National's support is also reflected in the poll question about whether the Government is moving in the right direction. The percentage answering "yes" dropped 5.5 points and the "no" response increased 5.9 points.
* The poll of 750 eligible voters was taken between September 17 and 23 and has a margin of error of plus or minus 3.6 per cent. The party vote result is of decided voters; 8.6 per cent of respondents were undecided.
- Audrey Young
