The museum hopes to raise £5m to develop its Maryfield tram depot into a fully-restored national visitor attraction.
It has been based at its temporary headquarters in Market Street since April 2014.
The grant will fund a new museum director and consultancy post for two years.
The museum's chairman Jimmy McDonell said: "We are extremely privileged to have been offered this grant from the Mathew Trust.
"It will allow us to carry the project forward by bringing in the expertise we need to make the very best of this opportunity that we have."
The museum, which is run by volunteers, attracted more than 10,000 visitors in its first year.
VisitScotland regional director Jim Clarkson said: "In Scotland's Year of Innovation, Architecture and Design, the grant will help Dundee Museum of Transport remain an exciting part of the city's tourism offering, boosting the visitor economy in the process."
Dundee Museum of Transport's fundraising campaign has received an £80,000 grant.