The Swindon forward stepped off the bench to grab goal eight of ten, but it proved to be crucial as Town held on despite late strikes by Matt Done and Che Evans to book a tie with Preston at the national stadium.
Mark Cooper’s side were 3-0 up inside 20 minutes, with Ben Gladwin scoring twice before Michael Smith’s third. But an own-goal by Nathan Thompson and Chris Basham’s header put the visitors back in the tie, only for Smith to grab his second of the night from the spot.
Steven Davies made things interesting again before Obika struck and then, despite two late strikes, Town eventually crept over the line.
The Blades made the trip 2-1 down on aggregate, and things could have been more comfortable for Swindon had Gladwin not missed a first-half penalty.
Within five minutes of the second leg, however, the midfielder had written those wrongs with a stunning opener, and Nathan Byrne’s refusal to give up on a lost cause down the right then earned Swindon a second. The full-back kept the ball in on the right and crossed for Gladwin to tap home.
The home support were then in dreamland with 18 minutes on the clock when Smith fired home, but the travelling Blades supporters were dreaming again two minutes later when Jamie Murphy’s left-footed shot deflected past Wes Foderingham.
United narrowed the deficit to just two goals seven minutes before the break when Bob Harris’s cross was met by the diving figure of Basham.
Jermaine Hylton then earned Swindon their fourth. Mark Howard was forced to foul the frontman as he dashed past him and strike partner Smith thumped home the penalty in the 58th minute.
Yet again, the visitors responded, substitute Davies heading home another Harris delivery at the back post just six minutes after his introduction to make it 4-3 on the night.
Goal No 8 arrived when substitute Obika picked up Massimo Luongo’s pass, cut inside and buried a low shot into the net. United were not done just yet, though. First, Done fired past Foderingham with two minutes to go and then substitute Evans cut inside and bent home a curling, left-footed shot to set up a nerve-shredding finale.