Alex McLeish's side came from behind to beat Israel 3-2 at Hampden Park on Tuesday, with James Forrest's hat-trick confirming their promotion and a place in the play-offs for Euro 2020.
Speaking to 'Sky Sports', Lambert remarked: "I think it's brilliant, I think the national team needed it and I'm glad the guys have done it because there has been so much negativity towards the national team and towards Alex.
"It's not nice and there been so many callers and that sort of thing and ex-players coming out and criticising the team when not long ago they were part of it themselves.
"Alex has done a great job in difficult circumstances to get through and I'm absolutely delighted."
He added: "We need to get to a major tournament, just for Scotland itself and for the cities, Glasgow, Edinburgh, Aberdeen, Dundee.
"I'm pretty sure if you go to these cities, if Scotland qualify, these places will be jam-packed. They'll be in the pubs, they'll be in the streets, just what England have had for the last few years.
"I'm delighted we've qualified and it gives us a chance. And, from myself, big congratulations for going through."