A first half dominated by the forwards ended with the hosts taking an 8-4 lead thanks to Carlos Tuimavave's late try.
But, within a minute of the restart, the French side led when Richie Myler scored after a flowing move from deep.
Liam Watts pounced to touch down to put the Black and Whites ahead again, but Walsh kicked Catalans to victory.
With both sides triumphant in their opening matches of the season, the victors were guaranteed to be early pacesetters after two games.
However, neither team played like leaders in a dour first period.
Catalans edged ahead 4-2 through two Walsh penalties, but the hosts led by four points when Tuimavave touched down following a fine burst by the outstanding Sika Manu.
Within 60 seconds of the restart the lead changed again as Australian forward Greg Bird combined with Thomas Bosc to tee up Myler for his second try of the season.
Watts then pounced from close range to restore Hull's advantage as the match opened up.
But some astute tactical kicking, and the reliable boot of Walsh, steered Dragons to just their second away over Hull in their past 13 attempts.
Hull FC coach Lee Radford: "It was painful to watch. The stop-start type of game was frustrating. The skillset 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."
Catalans Dragons coach Laurent Frayssinous: "I'm very pleased but at the same time it's only round two. 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."
Hull FC: Shaul: Fonua, Tuimavave, Griffin, Talanoa; Connor, Sneyd; Taylor, Houghton, Watts, Manu, Minichiello, Ellis.
Replacements: Green, Bowden, Washbrook, Thompson.
Sin Bin: Watts (67).
Catalans Dragons: Bosc; Broughton, Inu, Wiliame, Thornley; Walsh, Myler; Bird, Garcia, Horo, Moa, Aiton, Casty.
Replacements: Anderson, Bousquet, Baitieri, Da Costa.
Sin Bin: Bird (21).
Att: 13,544.
Ref: Robert Hicks (RFL).
Luke Walsh kicked 12 points as Catalans Dragons edged a bruising encounter with Hull FC to earn a second win in as many matches and go top of Super League.