San Antonio finished hitting 16-of-19 3-pointers.
But it was the exploits of Russell Westbrook who ended with 24 points, 13 rebounds and 24 assists.
San Antonio opened its 154-147 win by setting the National Basketball Association record for the most consecutive makes from three before a miss.
They head for Oklahoma City for a rematch Saturday night.
2006 was the last time that two teams combined for over 300 points in a game, not including the All-Star Game. The Spurs' Derrick White grabbed the rebound, and the game was all but over.More news: Jordan Peele Shares First Trailer for His Sci-Fi Comedy ‘Weird City’
Aldridge was 20 for 33 from the field and 16 for 16 on free throws in surpassing his previous career high of 45 points, which he set against Utah on March 23, 2018.
He scored a career-high 25 points, added 12 rebounds, and swatted three shots in his 44 minutes.
And that says nothing about the sheer drama we were treated to along the way. Let's hope not, but what the San Antonio Spurs and Oklahoma City Thunder did on Thursday night might be the landslide victor at season's end. San Antonio took a commanding 13-point lead into the fourth quarter after shooting 61 percent in the third. "Just body language. I felt like I had a good matchup and I was in a good rhythm out there".
All five Thunder starters scored 19 or more points with Paul George leading the way with 30.
The Spurs scored a season-high 70 points in the first half behind ideal 10-of-10 shooting from long distance - riding Aldridge and Marco Belinelli to a 70-63 advantage at the break.More news: 'It's not professional': Sarri blasts Bayern Munich over Hudson-Odoi
"Sometimes when it gets to clicking, you just see a different side of a player", guard DeMar DeRozan said.
The Threes. San Antonio started 14-of-14 from the three-point line and it appeared as though they'd run away with the game as a result.
Such a brilliant night for Aldridge seemed nearly inconceivable in November, when the veteran suffered through a shooting slump that saw him score six points on 3-of-12 shooting against the Phoenix Suns and 10 points on 3-of-15 shooting in 38 minutes against the Clippers.
DeRozan missed another jumper with 0.3 seconds left in the first extra period and the Thunder could not put up a shot at the buzzer as White tipped the inbounds pass from Westbrook.
As San Antonio looked to close things out in the fourth, his baseline turnarounds were too much for a depleted Oklahoma City frontcourt that was without backup center Nerlens Noel, and without Steven Adams for a stretch late, when he went to the locker room with an injury.More news: New survey shows Britons would now vote to stay in EU
Spurs: Aldridge has 17,758 career points, passing Magic Johnson for 78th on the NBA's career list.