The decision will have to be rubber-stamped at a later date once the developers make tweaks to their proposals.
But the outcome paves the way for the new "state of the art'" terminal, more flights and the expansion of daylight flying hours from 6am to 11.30pm.
Environmental campaigners and climate academics had warned flights in and out of Leeds needed to dramatically reduce in order to help humanity have a fighting chance of averting climate catastrophe in the coming years.
Leeds City Council said in a statement on Friday: "Current Government policy points to these emissions being something that should be primarily tackled at a national level – and addressed through international agreements and protocols – rather than by suppressing growth at individual airports in a way that could simply export passengers to other nearby airports at a higher financial cost to them and increase surface transport emissions."
GALBA said on their fundraising page: "On February 22, Leeds City Council approved Leeds Bradford Airport’s expansion plans.
"This was the wrong decision: wrong in law, wrong for our communities and wrong for our climate.
"The council has chosen to put itself on the wrong side of history.
"However this story is far from over - GALBA is mounting a legal challenge against the council’s decision and we urgently need to raise funds to pay for our barrister and other expert advisers. We need to raise £30,000."
Airport bosses want to increase the number of passengers at LBA from four million to seven million.
GALBA added: "That would mean the airport doubles its greenhouse gas emissions in the middle of the climate emergency. And it would mean much more noise, air pollution and traffic congestion affecting thousands of people in our communities.
"Local campaigns stopped expansion of Bristol and Stansted airports - we will do the same! Please make a donation today! All contributions, big or small, will help.
"You can donate by clicking on the Give Now button on this page. Thank you.
"We say yes! Yes to a safer, quieter, cleaner and greener future."
Regarding a possible legal challenge, LBA chairman Andy Clarke told the YEP on Friday that they would "face that when we come to it".
He added: "We've got so many supporters who want this to happen - we'll work with whatever is thrown at us."