Lawrence placed his shot inside Gabor Kiraly's left-hand post for his 14th goal of the season after Jon Harley had tripped Richard Cresswell.
Burnley got off to a cracking start with Kyle Lafferty giving them the lead from their first attack in the third minute.
A swift counter-attack involved former Stoke players Ade Akinbiyi and James O'Connor, before the unmarked Lafferty fired home from six yards.
Stoke looked momentarily stunned and Akinbiyi should have added a second when he mis-directed a close-range shot in the eighth minute.
A strong and swirling wind made conditions difficult and City struggled to build up a head of steam with a couple of wayward shots from Mamady Sidibe the best they could muster.
They finally threatened the Burnley goal a minute before the interval when Richard Cresswell's cross-cum-shot just evaded Lawrence's lunge at the far post.
Stoke had the elements more in their favour after the break and they attempted to up the ante as they attacked the Boothen End which was hosting their most vociferous fans.
Ricardo Fuller's shot was aimed too close to Kiraly and the Burnley defence did well to withstand a series of corners and the long throws of Rory Delap.
Stoke's assault continued and they finally created a real chance on the hour when Fuller latched onto Delap's pass only to guide his shot a foot wide of Kiraly's left-hand post.
Kiraly then did well to block a fierce free-kick from Danny Pugh and the visitors looked set to hold out until the late drama.