Puzzle Strike is related to Dominion in its buy-new-chips mechanics, but its attack and defense mechanics are more strongly tied to Super Puzzle Fighter II Turbo, the video game without a prequel. Puzzle Fighter has much of the same dynamic as Puzzle Strike: the closer you are to losing, the more ammunition you're holding.
I don't have any hard numbers to back me up about any first-player advantage, but this article about slippery slope vs. perpetual comeback (written by someone involved in the creation of both Puzzle Strike and Puzzle Fighter) puts Puzzle Fighter firmly on the "perpetual comeback" side of the scale: being close to death is also being close to launching a huge attack on your opponent. Winning isn't about slowly wearing down your opponent and forcing them to jump on a slippery slope of "losing = less resources" (such as in an attack-heavy game of Dominion). Instead, winning Puzzle Strike or Puzzle Fighter is about getting in the last, largest attack. Going first isn't quite irrelevant in that case, but winning is more about taking advantage of the current board state than buying cards one turn ahead of your opponents.