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The Dragons were reduced to 12 when Greg Bird was sin-binned after 21 minutes but two minutes later Walsh kicked them into a 4-2 lead.
Myler crossed a minute after the restart and two kicks from Walsh gave the French side a four-point lead, but Liam Watts ripped the ball away from Krisnan Inu and bundled over - Sneyd's conversion putting Hull back in front.
Walsh levelled after Taylor conceded another penalty and when Watts was yellow-carded for a dangerous tackle, Walsh was again on target for what proved the game-winning score.
Catalans Dragons coach Laurent Frayssinous insists his team will stay humble despite back-to-back wins over two of last year's top-four clubs.
"I'm very pleased but at the same time it's only round two," he said. "Something we learned from last year is to stay humble because we were in the top four at one stage and you saw how we finished.
"There are areas where we need to improve. We created some opportunities and could have scored more tries. Our last plays in the first half weren't good enough.
"But this was definitely a game we could have lost last season. We were patient, showed good defence and came up with the two points which is pleasing.
"The players work hard on and off the field and without a good team culture, you don't come to Hull FC and win, so it's a very good sign."
The Dragons' hard-fought win came at a cost with Jason Baitieri and Sam Moa sustaining ankle injuries, Julian Bousquet picking up a knee injury and Paul Aiton missing most of the second half with a suspected broken rib.
"I'm worried about injuries," Frayssinous said. "We had 14 players for the last 38 minutes. I need to look at the video to make sure Hull have the right tackle technique but there is some similarity with the injuries."
Hull coach Lee Radford admitted his side failed to create enough chances.
"It was painful to watch tonight," Radford said. "The stop-start type of game was frustrating. The skill set of the group wasn't where it needs to be.
"We didn't take any juice out of them at their end of the field.
"It was smart by them and it's not the first time they've done that. You can't keep giving the ball away.
"I thought physically we were as good as we've been for a long time but we've got to be better with the ball, we looked as bad as we've had with ball in hand."
Hull lost skipper Gareth Ellis early in the match with a neck injury which Radford confirmed as "a bit of whiplash".
This article originally appeared at rugby-league.com.