The 35-year-old Shevchenko, who battled nagging injuries to claim his place in the starting line-up, twice headed the ball past Andreas Isaksson to cancel out Zlatan Ibrahimovic's 52nd-minute strike.
"Every game is like a final for us," Shevchenko told reporters. "We showed our character and now have good chances to qualify from the group."
Ukraine, taking part in their first European Championship, moved top of Group D with three points, ahead of France and England who fought out a 1-1 draw in Donetsk.
Sweden looked on course for a routine win after a slow start when Ibrahimovic, only standing out up to that point by virtue of his white and pink boots, put his team in front.
Fielded in a free role as a playmaker, Ibrahimovic casually diverted a Kim Kallstrom pass into the net seven minutes into the second half.
Shevchenko, however, hit back three minutes later for Ukraine with a diving header from an Anatoliy Yarmolenko cross as a packed Olympic stadium burst into a huge roar.
"Sheva, Sheva!," chanted the crowd and they had not seen the last of their hero on an electric night.
The striker doubled his tally in the 61st minute when he headed home from a corner at the near post after losing his marker Ibrahimovic. He was later substituted and left the pitch to a standing ovation before being named man of the match.
Surprisingly it was Ukraine who imposed a possessive pattern in the early stages with neat quick passing around the box but the Swedish defence remained composed.
Sweden, however, looked nothing like the team who scored 31 goals in their qualifying campaign and almost fell behind midway through the first half just when they seemed set to gain control.
Shevchenko started a sharp counter-attack and was also on the end of it but, after being perfectly set up by Yarmolenko, he fired an angled shot narrowly wide as coach Oleg Blokhin held his head in disbelief.
Ibrahimovic was too laid back when his downward header bounced on to the far post just before the break and the miss was to proved costly, though substitute Johan Elmander missed a glorious chance to equalise in the last minute when he blazed the ball wildly over the crossbar.
"The first half wasn't good enough," said Sweden coach Erik Hamren.
"Both teams were very nervous, a lot of technical faults. We started the second half better, being more aggressive, winning the ball higher up. We put pressure on them, and that's what led to the chance where we scored.
"You also have an opponent who was really good - Shevchenko did really really well in the box, he was strong," Hamren added.
