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."

The Pilgrims have won nine of their last 10 league games and are five points clear at the top of League Two.
Last season's beaten play-off finalists beat Yeovil 4-1 on 1 October and have not lost in the league since 13 August.
"I've seen us in training and I've seen us perform and I still think we've got another gear in us," summer signing Bradley told BBC South West.
After last season's disappointment at Wembley, manager Derek Adams added 14 new signings and two more players on loan.
Former Crawley captain Bradley continued: "We're definitely capable of promotion.
"I've been part of different League Two squads up and down the country and I can tell you that if we play to our potential every week, we continue to listen to the manager and take on instructions, then this team is more than capable of automatic promotion.
"I wasn't here last season and don't have any reminders about Wembley and what happened.
"I don't care about that if I'm honest, all I care about is keeping clean sheets and making sure this club is getting where it needs to be. At the minute that's where we are - top of the table and on course for a successful season."

Handmade at Amazon is currently invitation-only, with an initial 5,000 sellers on the site.
Goods must be "factory-free", with items such as wooden cufflinks and handstitched baby bibs already on sale from around the world.
Its competitor, artisan website Etsy, was valued at $1.8bn (Â£1.2bn) when it launched on the stock market in April.
"We had thousands of searches every day from customers looking for handmade or handcrafted items,'' said Peter Faricy, vice-president for Amazon Marketplace.
"Knowing an item has a unique story behind it creates a personal experience that customers have told us makes owning handmade items special."
The firm will initially take a 12% sales fee from its handmade retailers with no other charges.
Etsy, which was founded 10 years ago, takes a 3.5% sales fee from its 1.5 million sellers and also charges a listing fee of $0.20 per item.
The firm's chief executive, Chad Dickerson, told the Associated Press that almost half of the site's sellers already used other retail outlets in addition to Etsy but that the site was still "usually" their main source of income.
"We believe that Etsy is the best platform for the creative entrepreneur," he said.
"Etsy has a decade of experience understanding the needs of artists and sellers and supporting them in ways that no other marketplace can."
Dundee Museum of Transport's fundraising campaign has received an £80,000 grant.
[NEXT_CONCEPT]
Plymouth Argyle defender Sonny Bradley says the League Two leaders still have better performances in them.
[NEXT_CONCEPT]
Amazon is entering the handicraft market with the launch of a new online store for handmade goods.