The Swans' display during the opening 45 minutes was out of the top drawer as they dismantled a West Brom side that had risen as high as third in the Premier League table.
Michu put the hosts ahead with his ninth goal of the season as he turned in Hernandez's pull-back, before Routledge found the net twice, seeing Jonas Olsson's attempted clearance hit his shin and fly in, before smashing home Ben Davies' cross to make it 3-0.
Romelu Lukaku pulled one back for the Baggies just before half- time but, despite an improved second-half display, that was as good as it got for Steve Clarke's side.
Swansea are up to eighth after a run of just one defeat in their last nine games, and Laudrup had nothing but praise for their display.
He said: "It was the best (performance) I have seen for several seasons. It is not often as a manager you can sit and enjoy your team playing.
"Always during a game there are moments when there are things you have to improve, but the first 45 minutes was outstanding, all 11 players played one-two touch football and everything was so good.
"It was one of the few times I can recall sitting there enjoying the game and it also showed everybody what we can do if we are at our best. It was great for me to see the players at their best.
"I will not sit here and say this is the best I have seen in my time but as a manager it is a pleasure to see a team play like that.
"I have not had that many times, maybe once or twice."
Having watched his side come up well short in their bid to secure a fifth-straight top-flight win for the first time in 34 years, West Brom boss Clarke had no complaints.
He said: "That was very unexpected.
"We started so slowly it was incredible, you wouldn't believe it.
"We know Swansea are a fantastic team and they play good football and we knew they would cause us problems, but I did not expect so many problems so early and when you are chasing the game it is going to be difficult.
"We have had a really good season, played really well. We came into the game confident, with more or less the same players and system, but that's football, it happens.
"They had a quick start, they got in front and we left ourselves too much to do to be honest."