Like Bubba Watson and Luke Donald before them, both men birdied to halve the first hole. However, the Englishman three-putted to concede a hole on the second green.
Poulter fell further behind at the fourth, when he found the bunker with his approach. He caught his rescue shot too well and flew out of the trap and onto the other side of the green.
Simpson, on the other hand, had dialled in a great shot at the flag and rolled in for a birdie.
Poulter halved the deficit with a birdie at the fifth, and was given the opportunity to square the match when Simpson fluffed his approach to the eighth.
The Englishman was able to two-putt for a par that saw him restore parity to the scoreline as they headed to the turn.