The bumper deal, which comes after talks between the pair were revealed in September, is set to create a combined business worth some £1.9 billion with a fleet of about 40,000 vehicles and a workforce of 70,000 people.
The deal, which has still to be voted on by shareholders, values Stagecoach at some £437 million.
About 50 roles are expected to be cut under plans to slash annual costs by at least £45m following the merger.
Those jobs are set to go across the head offices, IT and corporate departments of the two firms, as well as some overlapping senior management positions.
But the companies stressed there would be no front-line job losses, such as among drivers, or depot closures due to the tie-up.
In a bid to ease any potential regulatory competition concerns, Stagecoach also announced deals to offload the marketing, retail and customer service operations of Megabus UK and the Falcon South-West coach service, as well as its 35 per cent stake in the Scottish Citylink Coaches joint venture.
Stagecoach chairman Ray O'Toole will become the chairman of the merged group, while National Express chief executive Ignacio Garat will keep the same role at the enlarged business.
The merger comes as both firms have been hit hard by the pandemic, with passenger numbers slumping due to lockdowns, remote working and a switch away from public transport. Government support to help transport firms through the crisis is also due to end soon.
The deal follows a previous attempt at a merger in 2009, when National Express rejected a £1.7bn merger deal mooted by Stagecoach.
The former attempt at a marriage between the two would have seen Stagecoach own the majority of the combined group, with National Express left with up to 40 per cent.
Garat said: “The proposed combination of National Express and Stagecoach, and the unique strengths of both companies and their teams, will create a leading multi-modal passenger transport business in the UK.”
Martin Griffiths, chief executive of Stagecoach, added: “This is an exciting opportunity to bring together two of the UK's iconic transport brands to create a strong, diverse business that is well-placed to grow the market for greener and smarter public transport for the benefit of all stakeholders.”
National Express has bus and coach networks across the UK and Spain, while it also runs school bus services in America and a rail franchise in Germany. But Stagecoach is UK focused and is Britain's biggest bus and coach operator.
National Express said the deal will allow it to use Stagecoach's depot network to run and maintain its coach operations while also allowing it to expand its new growth initiatives - such as private coach hire, corporate shuttle and accessible transport - across Stagecoach's UK operations.