Motherwell looked to be coasting when Higdon swept home his third goal in the 69th minute after two early goals (eight and 20).
But Nigel Hasselbaink and Liam Craig (77 and 79 beat Lee Hollis within the space of two minutes just after the goalkeeper had replaced the injured Darren Randolph, setting up a nervy finish.
However, Hollis made an excellent late save from Hasselbaink to keep the three points at Fir Park and move the hosts level on points with second-placed Inverness Caledonian Thistle.
The match started in open fashion before Higdon struck in the eighth minute.
The former Falkirk and St Mirren striker set Chris Humphrey free down the right wing. Higdon was quick to support the winger and swept a wonderful first-time finish into the bottom corner from 16 yards after Humphrey had picked him out with a measured pass.
Home captain Keith Lasley cleared off the line after Craig's corner had glanced off the head of Motherwell defender Shaun Hutchinson and Stuart McCall's side moved two ahead in the 20th minute. Henrik Ojamaa rolled the ball into Higdon who failed to connect properly with his first effort after turning Frazer Wright, but the ball broke back to him 14 yards out and he fired a shot low into the net.
St Johnstone dominated the rest of the half as Motherwell struggled to get out of their own half with Ojamaa unable to get on the ball.
But Motherwell rediscovered their attacking drive after the break and went 3-0 up on 69. Hutchinson rose well to meet Tom Hateley's corner and his header broke off a defender to Higdon, who dispatched the ball into the corner from 10 yards.
Motherwell lost Randolph to what looked like a groin injury after MacLean had just failed to turn home Vine's low cross. Hollis was soon picking the ball out of the net after substitute Hasselbaink played the ball into MacLean and ran on to his lay-off to curl high into the corner from 18 yards.
Saints soon set up an exciting finish when Craig fired a powerful first-time effort into the bottom corner from 20 yards.
Hollis had little chance with either effort and he ensured the victory in the 89th minute when he rushed off his line to save from Hasselbaink after the Dutchman had forced his way beyond the home defence.