The 52-year-old Cook, an 11-time winner on the regular PGA Tour, finished at 11-under 205 on The Woodlands Country Club's Tournament Course.
Haas, bidding for a victory in his third consecutive start and one stroke off the lead at the start of the final round, shot a 71. He had three birdies on the back nine, but couldn't offset at early double bogey. He had late chances to tie, but a putt for birdie at the 17th lipped out.
At 18, his drive landed in the rough just short of the water guarding the green, then his shot to the green wound up long on the back fringe. When his long birdie attempt slid past the hole, Cook was the winner.
Tway, who shot a 70, started the rounds two shots off the lead, got close but never could catch the leaders. He had one bogey and three birdies, and closed with four straight pars. The eight-time winner on the PGA Tour joined the Champions Tour earlier this year after turning 50 and his previous best finish was a sixth-place tie a month ago at the Greater Hickory Classic in North Carolina.
Dan Forsman, the leader after the first and second rounds, was cruising Sunday and at 11 under through 13 holes. His run ended with a triple bogey at the par-3 14th, where he put his tee shot in the water. He finished with a 73 to tie for fourth at 8 under with 2007-08 winner Bernhard Langer (69) and Tom Lehman (68).
Cook was two shots behind Forsman when play began Sunday. Cook started birdie-bogey, but birdies at Nos. 5 and 9 got him within one shot of Forsman.
Cook got to 10 under, still a shot behind Forsman, with a birdie at 12, then claimed a share of the lead when Forsman bogeyed the hole. Cook took control after Forsman's triple bogey at 14 and pulled away with a birdie at 17.
Cook lost to Mike Reid in a playoff in August at the Tradition in Oregon, one of the Champions Tour majors. Cook was fifth two weeks ago at the Senior Players Championship, where he was in contention until fading in the final round.
