The 29-year-old, recalled to the Test team for the series after a two-year absence following the international retirement of Ben Stokes, struck an unbeaten 50 as England stretched their second-innings lead to 357 with three wickets left.
Lawrence struck two sixes and six fours in a 66-ball stay as England reached 7-177 when play was called off for the day.
Mohammad Abbas took three wickets in the bowler-friendly conditions to take his match haul to seven but even with his contribution Pakistan are in a hopeless position.
With two days left in the match, an uneven pitch and an improved weather forecast, particularly for Monday, England are odds-on to close out the win which would deliver a first series win since December 2024.
The target is almost certainly beyond Pakistan already. The tourists' fragile batting line-up has yet to score more than 171 in an innings in this series.
They were shot out for 110 in their first innings by England's potent pace attack and opener Imam-ul-Haq will be unable to bat after not taking the field today because of a calf injury.
England, who resumed on 3-81 when play eventually got underway at 2.20pm local time, lost Emilio Gay early, lbw to Abbas, after only adding a single to his overnight score.
Harry Brook pressed on to 34 before edging Abbas into the gloves of wicket-keeper Muhammad Rizwan, who took a good catch standing up the stumps.
First-innings half-century maker Jamie Smith fell for seven, chipping Abbas to Shan Masood at mid-wicket but Lawrence dug in.
He lost Gus Atkinson three balls after tea, lbw to Mohammad Ali, but took the attack to spinner Ali Usman, slog-sweeping him over mid-wicket for six to add to the earlier blow off Ali over the extra cover ropes.
A pulled single to deep mid-wicket off Ali brought up a deserved half-century.
It was his second in successive matches after making 51 in the first Test victory at Headingley that gave England a 1-0 lead in the three-match series.
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