Robbie Parry slotted the Welsh visitors into a half time lead as the rain lashed down in West Lothian.
Steven Saunders doubled the advantage from close range on 58 minutes before Quigley curled in a wonderful strike.
As Livingston toiled to make inroads in attack, TNS passed up chances to make their victory even more comfortable.
The runaway Welsh Premier League leaders, on their way to sixth successive title, now join Scottish Championship trio Dundee United, Queen of the South and St Mirren in Monday's draw.
TNS were content to sit back for long spells, with Scottish League One leaders Livi finding it hard to get a glimpse of goal.
When they did, Raffaele De Vita headed straight at goalkeeper Paul Harrison and then shot over from a tight angle before Liam Buchanan sent an angled strike narrowly wide.
The visitors went ahead on 36 minutes when Simon Spender cut in from the right and his blocked shot fell to Parry, who calmly picked his spot with a first-time finish.
Livi improved for a brief spell at the start of the second half, with Buchanan slamming a shot into the side-netting shortly after Harrison had done well to push away a strong header from De Vita.
TNS moved further ahead from a corner as the home side cleared off the line but the ball was sent back into the danger area and an unmarked Saunders scuffed home from inside the six yard box.
The contest was over when Quigley cut in from right and arced a sweet shot into the opposite top corner with his left from junction of the penalty box.
Late on, Christian Sergeant broke clear of the ragged Livi defence and looked certain to score but keeper Gary Maley got a slight touch on his chipped effort to slow the ball down, with Michael Miller sliding in to complete the clearance.
Maley had to look lively again, pushing over a rising shot from Parry as TNS ended the evening well on top.
TNS's Craig Harrison: "It was very tough physically but we matched them. Overall, it was a very professional performance and we were ruthless in the final third.
"We have a strong squad and we go into every competition wanting to win it. The whole club, from top to bottom, has that mentality. We want to win every game."
Livingston's David Hopkin: "We knew it would be tough with the run they are on but the goals we conceded were very poor.
"We huffed and puffed and had a few half chances but TNS deserve to go through."
Scott Quigley scored the pick of the goals as The New Saints brushed Livingston aside to reach the semi finals of the Scottish Challenge Cup.