Goals from left-back Ben Davies and the influential Pablo Hernandez made up for back-to-back defeats, albeit to Manchester United and Tottenham, to get the Swans' season up and running.
That sends Laudrup into the international break with a smile on his face and by the time Swansea play again, at home to Liverpool on September 16, he hopes to have another option up front.
Speaking ahead of transfer deadline day, he said: "I've been asked about a lot of players and I've always said we want one more striker with different characteristics to Wilfried Bony and Michu," he said.
"We will add one tomorrow. We've had five, six, seven players linked and one will join, but not more than one.
"We have to go into the last day (of the window) having not done everything which is not ideal because it will be a crazy day, but hopefully we'll finish in time.
"I don't want to go into details but let's say we hope to finish it as quickly as possible."
Laudrup was delighted to be leaving The Hawthorns with three points in the bank, admitting that his side's tough opening fixtures had placed real significance on the result here.
"We knew it was very important for us to get something out of this game," he said.
"When we saw the fixtures before start of the season - starting against the champions and then Tottenham away - we knew it was difficult. Then you look at squads in our league, as I call it, teams like West Brom and try to get something.
"It's not easy when you come here but it was a great win and great performance.
"It's always difficult to play away, wherever you play."
West Brom manager Steve Clarke included Nicolas Anelka in his starting XI less than a fortnight after reports emerged suggesting the French striker was set to retire from the sport.
The club denied he had ever decided to quit but did grant him compassionate leave following the death of his agent.
Anelka returned to training in midweek and Clarke had no qualms about giving him 90 minutes.
"I thought Nicolas' performance was really good and I picked him in the starting XI so I must have been happy with him (in training)," said the Scot, who appeared irked by questions about Anelka.
"Nicolas is part of the squad and he's a good player; his performance showed he is a good player. I think the fans like him, they like the fact that Nicolas Anelka plays for West Brom.
"Of course the crowd are behind him, he's one of their players. He's part of the squad and everyone is happy he's here."
Like Laudrup, Clarke is hoping to add attacking players on deadline day - with his need particularly pressing given his side's record of no goals in three league outings this season.
"I've spoken at length on about the need to bring fresh faces in to the club," he said.
"We have good people who have been working hard to make this club better and make them more competitive.
"I hope there will be some good news for the supporters, we all need good news...even I need some good news.
"I need some players in forward areas that can give us a bit of quality so we create chances."