Poulter made the ideal start with a birdie on the first and won the fourth and fifth with pars, McIlroy's drive on the latter hole going out of bounds and finishing up in a swimming pool.
McIlroy got back to one down with the aid of a superb birdie on the eighth, but Poulter won the next five holes in a row and McIlroy's miserable day was summed up when he chipped across the green and into the water on the 13th.
With the 64-man field split into 16 groups of four at Austin Country Club, McIlroy could still qualify for the knockout stages of an event he won in 2015.
But, given the manner of his performance on day one, it looks unlikely that the four-time major winner will be able to turn his game around sufficiently to beat Cameron Smith and Lanto Griffin, while Poulter knows wins over the same players will secure his place in the last 16.
Asked about his comprehensive victory, Poulter said: "I'm not sure it was easy, to be honest. I mean, you're playing Rory and it was apparent when you're one up after being three up that this is not going to be an easy match.
"And when he's kind of gifted me one on 10, you would expect him to have a half a chance to chip that in, you're not expecting him to not get up and down.
"Then I've hit a great shot into 11 and that's not an easy pin for him when he's moving the ball right to left. I could play my cut in there nicely and I just kept him under pressure.
"But I wanted to keep the foot down because it's too easy to let him get back in the game like he did and then he takes control of the match."
McIlroy has previously sought Cowen's input on an informal basis, including at the Players Championship earlier this month, but only when long-time coach Michael Bannon has not been at a tournament.
The 31-year-old Northern Irishman missed the cut in defence of his title at TPC Sawgrass after rounds of 79 and 75 and admitted that his swing had suffered after trying to add more length to his game, partly influenced by Bryson DeChambeau's dominant US Open triumph.