Two errors from home goalkeeper Tim Howard were punished as Leicester responded to Steven Naismith's opener with goals from David Nugent and Esteban Cambiasso.
The Premier League's bottom side looked set to climb within two points of safety as a result, only to suffer late heartbreak as Upson inadvertently diverted a Romelu Lukaku header into his own net.
Naismith put Everton in front after 57 minutes with a scuffed finish, but the game then turned on a brace of mistakes from the experienced Howard.
The United States shot-stopper spilled a low cross to gift boyhood Everton fan Nugent a 63rd-minute equaliser and failed to deal with Danny Simpson's right-wing delivery in the build-up to Cambiasso's close-range strike seven minutes later.
Lukaku spurned a host of chances for Everton, but the Belgium forward ultimately came good as he met Christian Atsu's cross and saw his diving header brush off Upson to leave Mark Schwarzer with no chance.
Everton saw left-back Leighton Baines return to the side after three games out with a knee problem, but his recall did not inspire a fast start as both teams offered little going forward in the opening stages.
Lukaku perhaps should have given Everton the lead in the 22nd minute, only to blaze over the crossbar from six yards following good work down the right-hand side from Seamus Coleman.
In-form lone frontman Andrej Kramaric then went close for Leicester, who recalled Robert Huth, the Croatian curling over from the edge of the area.
Jeffrey Schlupp may have broken the deadlock with the final kick of the first period had John Stones not produced a fine block to deflect his effort wide with the goal gaping.
But if Schlupp was unfortunate not to score on that occasion, he had no excuse just after the restart as a gilt-edged chance went begging.
The midfielder somehow skewed wide of the right-hand post on the rebound after Howard had only managed to parry a low effort from Matty James.
Leicester were made to pay for Schlupp's profligacy when Naismith struck his sixth league goal of the campaign.
Naismith collected the ball from Lukaku and worked his way into the box before squeaking the ball just inside the near post.
Everton, seeking a first home league win since mid-December, were subsequently pegged back as Howard failed to collect a low cross from Jamie Vardy, with the ball the ricocheting off Stones and into the path of Nugent to tap home.
Lukaku then curled over when through on goal at the other end and Leicester took advantage by capitalising on another Howard gaffe to complete the turnaround.
Howard flapped at Simpson's lofted cross, before Vardy pulled the ball back for Cambiasso to turn home.
Cambiasso's strike looked to have ensured a vital three points for Nigel Pearson's men, but the unfortunate Upson inadvertently levelled matters again before Kramaric and Lukaku spurned presentable chances in a thrilling finale.
