Anderson, who started the fourth day’s play of the third Test with 598 wickets at this level, induced an edge from Shan Masood, then on three, but wicketkeeper Jos Buttler floored a seemingly routine chance.
It was the fourth dropped catch in 37 balls off Anderson and the closest the hosts came to a wicket in the session.
Pakistan were 41 without loss after being made to follow-on when rain stopped play at 12:21 pm local time (1121 GMT) and saw the umpires take an early lunch.
Masood was 13 not out and Abid Ali 22 not out, with Pakistan still 269 runs adrift of England’s mammoth first innings 583-8 declared.
England are 1-0 up in the three-match contest and will win their first Test series against Pakistan in a decade if they avoid defeat in Southampton.
No paceman has yet reached the landmark figure of 600 Test wickets, with the only bowlers ahead of the 38-year-old Anderson in the all-time list a trio of retired spinners — Sri Lanka’s Muttiah Muralitharan (800 wickets), Australia’s Shane Warne (708) and India’s Anil Kumble (619).
The 38-year-old Anderson started the day in sight of the landmark after taking 5-56 — his 29th five-wicket haul in 156 Tests — during Pakistan’s first innings 273.
Anderson might even have got there on late on Sunday had not three catches been dropped off his bowling with the new ball.
And there was more frustration for the swing specialist when left-hander Masood edged an intended drive outside off stump only for the ball to hit Buttler on the thigh as the wicketkeeper failed to get his gloves to a chance far less difficult than several catches he held on Sunday.
Before Pakistan could begin their second innings on Sunday, the umpires ruled the light was too poor for play to continue, even though the floodlights were on, and ended play for the day.
Pakistan resumed Monday a huge 310 runs behind an England total built on Zak Crawley’s 267 — the 22-year-old batsman’s maiden Test century — and Buttler’s 152.
Pakistan captain Azhar Ali’s superb 141 not out kept England at bay longer than looked likely after his side collapsed to 75-5 on Sunday.
Azhar, however, found an ally in Mohammad Rizwan, the wicketkeeper making 53 in sixth-wicket stand of 138 with his skipper.
This was just Azhar’s third score over fifty in 14 innings, but during his hundred he became only the fifth Pakistan batsman to have scored 6,000 Test runs.