But for a magnificant display by skipper Steven Gerrard, who grabbed his first career hat-trick, the Welsh Premier League champions could have come away with an even closer result.
The mission impossible started to take shape after Gerrard netted twice in the opening 21 minutes.
First Riise fired in a low cross which Anthony Le Tallec flicked onto the far post and Fernando Morientes found Gerrard to slot in from 10 yards.
Then, as the European champions continued to dominate, the midfielder forced his way past Steve Evans to lob in his and Liverpool's second to all but seal the win.
However, as TNS began to become familiar with their surroundings, they managed to play some neat football of their own while holding Liverpool at bay for just over an hour.
Evans recovered well to lead the defensive effort, while goalkeeper Gerard Doherty put memories of a dreadful performance at the City of Manchester stadium two years ago to the back of his mind to make some inspired saves from Liverpool's world class players including Gerrard and Morientes.
Ken McKenna would have been more than happy to take just a two-goal deficit to Wrexham's Racecourse Ground on Tuesday for the second leg, but a lapse of concentration allowed Gerrard to capitalise and fire in a 25-yard special for his hat-trick.
McKenna mused: "Maybe we can also come back from 3-0 down, just like Liverpool did against AC Milan.
"But seriously, I felt the third goal was a killer. We were so close to an excellent result. But I was still delighted with a magnificent display.
"We had a gameplan and stuck to it. A couple of individual mistakes cost us.
"But in the end it was Gerrard who won it. He is unbelievable. He played against us just like he did in the Champions League final, and that's a credit to him.
"You can't stop people with that quality."