The Australian duo had led for most of the match and were never behind until Leishman's 10-foot birdie chance at the final hole slipped wide.
Mickelson has played in all 12 Presidents Cups since the event was created in 1994 and improved his career record to 24 wins, 16 losses and 13 halves.
Woods, a U.S. assistant captain at the event, has a 24-15-1 career record.
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"I get that I can't dance and I get that I can't take selfies, but I can putt," Mickelson, 47, quipped in reference to a bungled attempt to take a photo with U.S. Presidents Barack Obama, George W. Bush and Bill Clinton on Thursday, and an awkward celebration dance with Kisner on Friday.
"I don't really think much about," Mickelson added of matching the record.
"I just love every opportunity to be a part of these events and on these teams, and this team is a unique team from any in the past in that the talent level is high or higher than we've ever had."
U.S. captain Steve Stricker said Mickelson had been upset at the putt he missed for the win on Thursday.
"I know he probably lost a little sleep over that last n night," Stricker said.
"He brought it up to me again this morning. So it was eating at him. He just (could not) believe that he missed that putt yesterday.
"It's nice to see that he got a little redemption."
(Reporting by Andrew Both; Editing by Greg Stutchbury)
