The 37-year-old, who retires from Formula One at the end of the season, recorded 1min 39.957 sec to edge out newly-crowned quadruple world champion team-mate German Sebastian Vettel.
Webber, who will start from the front of the grid for only the second time this season, equalled the Australian record of 13 poles set by Alan Jones, who was celebrating his 67th birthday on Saturday.
In a thrilling finale to an intriguing day, the two Red Bull men crossed the line after the chequered flag to clock their best laps, but battling Briton Lewis Hamilton failed to do the same as he spun his Mercedes in the final part of his lap.
“Fantastic guys, thank you for a fantastic job,” said Webber on the Red Bull team radio. This weekend has gone pretty smoothly,” he added later.
“Not the smoothest path to the start of Q2 or Q1, we were not electric, but we changed the car quite a bit and I got more and more comfortable as the session went on.
“It made me more comfortable in the final sector, where Seb has been quick, so I was able to match him there.”