Eder, playing in his first major finals for Italy after being naturalized last year, came close to being substituted by Antonio Conte after a dismal opening half up front alongside Southampton striker Graziano Pelle. But the Inter Milan striker had the Italy bench celebrating wildly on the pitch at the Stadium de Toulouse with a well-taken 88th minute strike after running on to Simone Zaza's header to beat Andreas Eriksson at his far post.
“I'm very happy with the goal,” said Eder, who struggled to make an impact after moving from Sampdoria to Inter in January.”I'm delighted for the team, and I'm very happy the coach has kept faith with me.
“I think it helped me I wasn't playing for Inter Milan a lot at the end of the season. That helped me to save some energy for this championships and also allowed me to work a lot of my fitness.”
Italy, 2-0 winners over Belgium last week, now top Group E with six points, leaving Sweden, who had late claims for a penalty waved away, realistically needing to beat Belgium on Wednesday if they are to have any hope of making it to the knockout phase.
After a mediocre display in Sweden's 1-1 draw with the Republic of Ireland, all eyes were on captain Zlatan Ibrahimovic as the towering striker sought to make his mark by becoming the first man to score in four editions of the competition.
But a cautious Italy restricted the former Juventus and Paris Saint Germain striker to few real chances in front of a huge army of yellow-clad Sweden fans.
A tight first half saw few chances at either end as Italy's formidable three-man defence of Andre Barzagli, Leonardo Bonucci and Giorgio Chiellini kept Ibrahimovic in check, whilst Pelle and Eder failed to shine on their few real forays up the park.
Veteran Italy captain Gianluigi Buffon had little to do in goal until he dived to collect Kim Kallstrom's curling shot, and the Juventus 'keeper was happy to see Olsson's drive whistle past the post soon after.
Sweden edged closer as the half wore on, Sebastian Larsson chesting a long cross into the path of Celta Vigo's John Guidetti only for the striker to skew wide.
Italy started the second half in positive fashion, Pelle controlling Eder's cutback from a Marco Parolo through ball, but his volley dipped over the crossbar.
Parolo did well to control De Rossi's pass and set up Candreva on the right, but the Lazio man's drive into the area was collected by Isaksson, who got down quickly to smother moments later after another Candreva delivery.
On that occasion, Eder was well out of position but it was Pelle who was replaced on the hour, by Zaza.
Minutes later, Florenzi's high delivery across goal found Candreva at the back post, yet Isaksson again got down low to smother the Lazio man's low first-time effort.
Italian hopes looked to be dying out and the Azzurri breathed a sigh of relief when Ibrahimovic fired over from a yard out at the back post although he was ruled offside.
Parolo was inches away from the opener on 82 minutes with a header that came off the woodwork.
But it was Eder who rescued Conte's men with a fine run that took him past the Swedish defence to fire the ball into the net with two minutes to play.