The Galaxy defeated Houston Dynamo 1-0 thanks to Landon Donovan's 72nd minute winner but Arena reserved his superlatives for Beckham who overcame illness and a hamstring injury to play a major role in the victory.
"David's a champion," the Galaxy coach said.
"I've been around great athletes and competitors in my life in different sports and this guy is as good as they come - unbelievable desire to win and compete.
"He gutted it out tonight. He obviously wanted to be there.
"My hat is off to David."
Beckham is now out of contract with the MLS champions and Arena said he would back him in whichever path he chooses to take regarding his future at the Galaxy.
"Whatever he decides to do, I support it 100 percent - he's earned that," added Arena. "If he decides to get on his horse and ride into the sunset or go elsewhere or retire, I support that.
"He's earned it. He's really earned the right to make the kind of decision he wants to make."