Fresh downpours meant play did not get under way on the last day until 1515 GMT but the 38-year-old Anderson struck with his 14th ball of the day to reach the milestone.
Defying a docile pitch, he produced a rising delivery that moved away from first-innings century-maker Azhar on 31, with England skipper Joe Root holding a head-high catch at first slip.
There was applause and cheers from Anderson’s team-mates, with the nick off the shoulder of Azhar’s bat clearly audible at an empty Ageas Bowl.
Azhar’s exit left Pakistan, following on, 109-3 — still 201 runs adrift of England’s first innings 583-8 declared, with the match heading for a draw after losing hours to bad weather.
England, 1-0 up in the three-match contest, are on the brink of a first series win over Pakistan in a decade.
Anderson is only the fourth bowler — and first paceman — to reach the mark of 600 Test wickets.
The only men ahead of him in the all-time Test list are three retired spinners — Sri Lanka’s Muttiah Muralitharan (800), Australia’s Shane Warne (708) and India’s Anil Kumble (619).
Kumble tweeted: “Congratulations @jimmy9 on your 600 wickets! Massive effort from a great fast bowler. Welcome to the club.”
Former Australia paceman Glenn McGrath, whose tally of 563 wickets was passed by Anderson in 2018, told the BBC: “Absolutely incredible, I’m a big fan of Jimmy’s. Just the fact he’s still playing now, in his 156th Test match, which in itself is just incredible.
“The rigours, the stresses that fast bowlers put on their body day in day out, to turn around and front up and just work your backside off is just absolutely incredible.”
Anderson was left just one wicket shy of the landmark figure as a combination of poor catching, bad weather and determined Pakistan batting thwarted him on Monday.
Denied an early breakthrough when Jos Buttler dropped a routine chance to reprieve Shan Masood, he returned late in the day to have opener Abid Ali lbw for 42.
But that was as good as it got for Anderson, who took 5-56 — his 29th five-wicket haul in 156 Tests — during Pakistan’s first-innings 273.
He might even have reached the landmark late on Sunday but he was frustrated by a clutch of dropped chances.
One of those at fault was long-time pace partner Stuart Broad, who paid tribute to his teammate on Tuesday.
“You can’t really put it into words, it’s just phenomenal,” he said.
“It doesn’t feel two minutes ago he broke Glenn McGrath’s record at The Oval against India with such scenes of taking the final wicket to win the Test match.
“He has got better with age and is someone who has inspired me throughout my career, watching him. The last five years in particular, since leaving South Africa in 2016, he’s just gone from strength to strength and he’s a role model to follow for every English cricketer and young cricketer coming through.”
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