With the teams above them getting positive results over the weekend, it was a crucial win for the Black Cats as they target automatic promotion.
But Hysen insisted Sunderland are the cool Cats of the race for the Premiership.
He said: "As long as we keep on winning then the pressure is on the teams above us. We have been playing catch-up for a long time but we're really setting our sights on the top two.
"It is up to Derby and Birmingham to protect their places and they'll know that we're on such a great run that we will give them a problem."
Hysen appeared as a first-half substitute after Carlos Edwards was taken out of the game by a heavy Richard Chaplow challenge, which went unpunished.
But the change of personnel did not knock Sunderland out of their stride as goals from Trinidad & Tobago internationals Dwight Yorke and Stern John put the visitors in command at The Hawthorns.
Hysen said: "It was obviously a very important win. West Brom would have gone six points ahead of us if they'd won, but now we're equal on points. It keeps our good run going and hopefully we can carry on from here.
"I think everyone's pretty tired and the pitch down there was heavy from the beginning. It was hard, but now we have closed the gap between ourselves and West Brom."
The Wearsiders' travelling fans were in fine voice again, and there will be almost 8,000 of them at Barnsley next weekend.
Swedish winger Hysen can't wait for the next game.
He said: "Our fans are always great and I'm just happy we were able to reward their support. We will love it at Barnsley with them behind us and hopefully we can continue this good run of away results for them."