Labour leader Andrew Little said it was the Greens' decision. "I welcome it."
He said it did not mean Labour now owed the Greens and there had been no discussions on any other electorates as yet.
Turei said it showed the memorandum of understanding was working. The Green Party reached its decision after several weeks of discussion.
"The decision was made by the Green Party, and has no bearing on any other electorates or the 2017 election campaign."
In the past Labour and the Greens have criticised National for doing deals to secure electorate seats for Act and United Future.
The byelection provides an early test of whether there will be any backlash from their own supporters by doing similar deals.
The announcement got a swift reaction from the Act Party on Twitter. It accused Labour and the Greens of "absolute hypocrisy."
"In Epsom, no party ever pulled a candidate yet it was still a scam according to Opposition. Now they go further in Mt Roskill - shameless."
Wood was Labour's candidate in Epsom in 2014 and described the deal between National and Act leader David Seymour as a "rort" and "a distasteful manipulation of the electoral system."
"ACT has turned Epsom into New Zealand's 'rotten borough' and this needs to end," Wood said at the time.
Seymour said it appeared Wood was either a "slow learner" who had just realised how MMP worked or lacked character.
"It will be interesting to hear whether he now thinks Mt Roskill is a rotten borough."
Wood's stunts included taking a sack of flour to debates as a fill-in for National's candidate Paul Goldsmith.
The Act Party also tweeted a photo of a bag of quinoa, saying "Hey @michaelwoodnz, we can't wait for your use of props in the Mt Roskill debates."
Although Prime Minister John Key had urged National voters to vote for Seymour and United Future leader Peter Dunne in Ohariu, National had still stood candidates in those seats.
The Mt Roskill deal could prove critical - in 2014 Goff won with an 8,000 vote majority over National's candidate Parmjeet Parmar but National got 14,275 party votes - about 2000 more than Labour.
The byelection will test how much of Goff's support was down to incumbency.
The Green Party candidate Barry Coates got 1682 votes and the Greens got 3279 party votes.
Coates will enter Parliament next week to replace Kevin Hague on the list.
Goff has said he will resign from Parliament immediately if elected Auckland Mayor early next month, allowing for a byelection before Christmas.