Playing their second overtime game in five days, the Texans improved to a league best 10-1 when Graham redeemed himself by nailing the clutch kick with 2:21 to play in the extra session, just minutes after missing one from 53 yards.
"We were tested physically big time," Texans coach Gary Kubiak said. "We've played 10 quarters of football in five days. I mean, that's just unheard of and we got beat up, too, so it was really a test of our character.
"I don't think that I've ever been a part of playing that much football in such a short period of time."
Shortly before Graham's winner, the Lions (4-7) had a chance to snap their nine-year Thanksgiving Day losing skid but watched in horror as Jason Hanson's 47-yard attempt clanged off the right upright.
The Lions and Texans got the traditional US Thanksgiving Day football feast off to a wild start as the two teams combined for more than 1,000 yards in total offence.
Texans quarterback Matt Schaub, who passed for 527 yards to tie for the second-most in an NFL game in an overtime win against the Jacksonville Jaguars on Sunday, did not come close to matching that performance, completing 29-of-48 attempts for 315 yards, including a nine-yard touchdown to Owen Daniels.
Arian Foster ran for 102 yards and a pair of touchdowns but his backfield partner Justin Forsett was in the spotlight after scoring what is sure to go down as one of the most bizarre and controversial touchdowns of the season.
Trailing the Lions 24-14 in the second quarter, Forsett appeared to be tackled on Houston's 25 but he spun up off the turf and race unchallenged into the Detroit end zone as players from both teams stood and watched.
The 81-yard touchdown run was the longest ever for the Texans franchise.
Lions coach Jim Schwartz then made a fatal error, throwing the challenge flag to cancel out an automatic review of the play under an obscure rule.
"I know that we can't challenge a turnover or a scoring play and I overreacted," Schwartz said. "I was so mad that they didn't call him down - because he was obviously down on the field.
"I had the flag out of my pocket before he even scored the touchdown and that's all my fault.
"I overreacted in that situation and I cost us a touchdown."
The Lions got the Thanksgiving Day festivities off to a rousing start, taking the opening kickoff and marching 80 yards with Mikel Leshoure capping off the drive with a two-yard run, the first rushing touchdown allowed by the Texans all season.
It also marked the first time this season the Lions had scored on their first possession.
Houston answered in the second quarter with Foster romping six yards for his 11th rushing touchdown of the campaign.
But for the rest of second quarter the Lions and Texans turned to an air attack that produced three touchdowns in just over four minutes, Stafford tossing strikes to Mike Thomas and Calvin Johnson while Schaub found Daniels in the end zone to cut the Detroit advantage to 21-14 heading into the intermission.
Detroit was on the scoreboard first again in the second half, Hanson connecting on a 46-yard field goal but the Texans came storming back with Forsett's controversial touchdown and a 45-yard Graham field goal to make it a tied game.
The fired-up Lions responded with a four-play quick strike touchdown drive, Joique Bell racing the final 23 yards for the score to put Detroit back on top 31-24 but the Texans would force overtime with a 97-yard drive capped by a one-yard run from Foster.
"We were so hungry at the end there to get a victory after battling all day against a good football team in their place," said Schaub.
"We played about as much football as they'll let you play within five days and we're just thankful that we came away with a win."