Wilder had Fury on the canvas twice, including a spectacular final round knockdown, but was unable to get the knockout victory he had promised to deliver at the Staples Center.
The three judges were divided on the outcome, with one scoring it 115-111 for Wilder, another 114-112 for Fury and the third 113-113.
"I think with the two knockdowns I definitely won the fight," said Wilder afterwards.
"We poured our hearts out tonight. We're both warriors, but with those two drops I think I won the fight."
Wilder, who remains unbeaten after 41 fights, immediately called for a rematch.
"I would love for it to be my next fight," Wilder said.
"Let's give the fans what they want to see. It was a great fight and let's do it again."
Fury meanwhile insisted he had done enough to win.
"We're on away soil, I got knocked down twice, but I still believe I won that fight," Fury said.
"That man is a fearsome puncher and I was able to avoid that. The world knows I won the fight."
Fury also said he hopes to arrange a rematch.
"One hundred percent we'll do the rematch," Fury said.
"We are two great champions. Me and this man are the two best heavyweights on the planet."
Until a sensational final round knockdown from Wilder, Fury appeared to be heading towards what would have been a remarkable upset.
The 30-year-old "Gypsy King", who returned to boxing this year after missing more than two years through depression, drink and drug problems, had boxed cleverly to evade the heavy-hitting threat of Wilder for most of the fight.
The American champion struggled to connect cleanly with Fury throughout an absorbing contest, all too often sending huge arcing haymakers whistling past Fury's head.
Wilder landed just 71 of 430 punches thrown, or 17 percent.
Fury by contrast cleverly picked his moments, finding Wilder with greater accuracy and causing a nasty swelling over the American's left eye. Wilder, the more aggressive of the two fighters early on, quickly moved into an early lead.
But Fury gradually grew in confidence, regularly taunting Wilder by throwing his arms up in the air or behind his back.
Wilder responded with increasingly desperate flurries of big punches, very few of which found their mark.
As several former heavyweights had predicted beforehand, the longer the fight went on, the more Fury looked in control.
- Fury goes down -
However in the ninth round, Wilder finally made his mark, dropping Fury with a short hook that had the fans on their feet.
Fury recovered well however and regained his composure to resume where he had left off. With a large contingent of British fans in the audience of 17,698 roaring him on, a victory suddenly seemed within reach.
Wilder however suddenly found a devastating combination just when he needed it in the 12th round.
A right hand sent Fury rocking backwards towards the deck and a brutal left hand on the way down appeared to be the coup de grace for the challenger.
Incredibly however, Fury managed to pick himself up and clear his head to survive the remainder of the round and escape with a draw.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)