The City skipper completed a pulsating nine-minute fightback with a bullet header to keep the Potters promotion dream alive and licking.
The City's hopes of a top spot looked buried when former Potter Chris Iwelumo fired in a dubious penalty, but City rolled out the second-half cavalry to virtually flatten the U's own play-off hopes.
Darel Russell and Mamady Sidibe earned the Potters a deserved lead, before Higginbotham finished it off to send City within touching distance of the Promised Land.
The nerves in the City camp were there for all to see during the opening sparring and the visitors should have taken advantage, both Jamie Cureton and Iwelumo asking questions.
Fortunately for the Potters the safe hands of Steve Simonsen had all the answers for the U's duo early on, tipping over and palming wide in quick succession.
The hosts did manage to put together a period of sustained pressure and they should have claimed a bounty when Lee Hendrie got on the end of Ricardo Fuller's power-play.
But the on-loan midfielder somehow lifted his effort over the bar before Fuller himself managed to slam a point-blank effort straight at the advancing frame of Dean Gerken.
And the hosts were left cursing their lack of composure in the danger zone when referee Graham Salisbury spotted an infringement in Salif Diao's challenge on Iwelumo.
But as Stoke's protests raged at the harsh decision the towering front-man dusted himself off, before firing his side into the lead from the resulting penalty kick.
A subdued atmosphere settled on the Britannia Stadium at half-time - it was to be the calm before the storm.
The big game jitters had blighted the standard of football in the first half, but the shackles were off and City took up a relentless assault of the U's goal.
Sidibe managed to nod back Lawrence's over-hit centre for Russell, who steered a towering header beyond Gerken's out-stretched frame to get the Potters pumping.
And with parity restored and the momentum swung, Stoke piled forward and it wasn't long before they had their lead when Sidibe latched on to Lawrence's through-ball.
The energetic target-man showed a touch of class on the 57th minute to pick out the bottom right corner from distance to set up a grandstand finale for his captain.
And Higginbotham powered in Lawrence's corner to move Stoke into a joint sixth spot, before the U's Chris Barker saw red for a raised arm on Hendrie to conclude a day which started with such promise for the visitors.