After shipping five goals in their last two home fixtures, Higgin’s penalty early in the game, after Joe Gardner was adjudged to have been fouled in the box could have sent an air of inevitability around Hartsdown Park.
Margate didn’t let that damage their belief though as Adeboyejo tested Lukas Lidakevicius from the edge of the box.
Kane Rowland saw shots blocked and skim wide in quick succession, with Harry Phipps and Jermaine Osei also narrowly missing the target for the hosts soon after.
On the half hour mark, Margate squandered a big chance when Adeboyejo was played through on goal but saw his effort half saved by Lidakevicius with Rowland thwarted by a recovering defender on the rebound.
Debutant goalkeeper Valerie Pajetat then kept his side in the game either side of half-time. Firstly he saved brilliantly from Gardner, with fellow debutant Cameron Heffernan clearing from under his crossbar and then he stuck out a big hand to thwart Gardner again from a deflected effort.
Those saves proved crucial as Adeboyejo won a penalty minutes later, burying an unstoppable effort into the top corner for a debut to remember.
There was only one team that looked like winning from then on, with Adeboyejo blasting a shot over from the angle and substitute Keanu Williams seeing a fierce shot from range well held by Lidakevicius.
Margate were so close to sealing their place in the next round of the FA Trophy in the 94th minute, only to see his half-volley from 20 yards skim just over the bar.
Margate: Pajetat, Heffernan, Tumwa, Phipps, Foster, Johnson, Richefond, Osei (Barrington 78), Adeboyejo, Rowland (Williams 61), Banya. Subs not used: Kallmeier, Bull.
East Thurrock United: Lidakevicius, Burns (Walker 59), Sammons, Clark, Wood, Ferdinand, Harris, Marlow, Higgins, Gardner (Craddock 78), Wraight. Subs not used: Honesty, Freiter.
Man of the Match: Alfie Foster (Margate)