The League Two newcomers fell behind on 27 minutes when Joshua Mullin headed in a free-kick from Nicky Cadden.
Mullin then provided a neat cross for Daniel Mullen to knock in right on the stroke of half time.
Mikey Miller crashed a shot against the woodwork as the visitors from League One continued to dominate before Liam Buchanan made it 3-0 from close range.
Ousman See went close for the hosts but was denied a landmark goal by a smart save from Liam Kelly.
Livingston move up to second place in Group G, behind St Mirren, who have two wins from two.

Asked by a journalist about reports on the personal wealth he has accumulated since leaving office, he said these had been "greatly exaggerated".
"I haven't actually changed, despite what people want to say," he said.
"The same thing that motivated me when I stood here... 20 years ago motivates me today. It's not about making money, it's about making a difference."
Mr Blair said his job as a Middle East peace envoy was unpaid, and he said that in total he spent two thirds of his time on unpaid work.
But his businesses supported 200 jobs, and needed to stay afloat, he added.
"I read that I am supposed to be worth Â£100m," the former prime minister said in a question-and-answer session after he delivered the Philip Gould Lecture to the New Labour think tank Progress.
"Cherie is kind of asking where it is. I'm not worth a half of that, a third of that, a quarter of that, a fifth of that, and I could go on."
His two businesses relied upon "a whole infrastructure, and I need to support that", he continued.
He said that "despite what people want to say about how I've changed" he still had the same motivation "but I'm doing it today as a private citizen, and I'm doing it in some of the most difficult places in the world, but I'm doing it with the same passion and purpose as ever I had before".
In a wide-ranging speech, Mr Blair said "radical" policies that bridged the divide between left and right-wing traditions were needed to propel the Labour Party to victory under Ed Miliband.
But he praised his successor, Mr Miliband's judgement on the EU. Mr Miliband has said if he wins the 2015 general election there will only be an in/out referendum on the EU if there is a significant transfer of powers proposed from Westminster to the EU institutions.
Mr Blair said: "I support Ed's leadership very much on this issue. I think he has made the right call on this and I think it is important to give him credit for that."
Two decades to the day since Mr Blair won the leadership of the Labour Party, he defended his record in government.
"We realised that there was no possibility of the British people accepting a programme which either pivoted on more state control or which didn't open up power for people as individuals," he said.
"The same is true today. That centre can and should be radical, but it will cut across traditional right-left lines because, over the past 100 years or so, people have learned from their experience about the state and the market that neither, untrammelled, is the route to the future.
"There will inevitably be a partnership between the two. And there will be a host of new ideas around how each might work and in combination with each other and the voluntary sector."
He also said he was "saddened and angry about the tragedy of the Gaza conflict".
Mr Blair, who is now a Middle East envoy working for the US, Russia, the UN and the EU, said the people of Gaza were "pawns in a wider struggle".
"Hundreds of Palestinians, including many children, [are] losing their lives in this nightmare," he continued.
"And I've witnessed the fear and insecurity of Israelis who still see their right to exist under challenge.
"I long to see two peoples and two states side by side in peace."
Edinburgh City's first competitive match as an SPFL side ended in League Cup defeat to Livingston at Meadowbank.
[NEXT_CONCEPT]
Tony Blair has said he is as motivated by "making a difference" as he was when he became Labour leader 20 years ago.