Former Reading and Chelsea midfielder Sidwell blasted a shot from the edge of the penalty area past Tim Howard after only 34 seconds following an error by Mikel Arteta.
Although Everton battled back to equalise through Joleon Lescott, a dreadful Phil Jagielka error allowed Ashley Young to restore Villa's lead.
Lescott scored an acrobatic stoppage-time equaliser but, straight from the restart, Ashley Young burst clear to snatch a dramatic winner for Martin O'Neill's side.
Everton boss David Moyes was without injured strike trio Yakubu, Louis Saha and James Vaughan so was forced to start with only Victor Anichebe in attack.
He also had to name youngsters Lukas Jutkiewicz, Dan Gosling and John Paul Kissock on the substitute's bench such were his lack of options.
Villa's amazing start immediately put Everton on the back foot and O'Neill's side could easily have doubled their lead with Martin Laursen's header from Ashley Young's corner being deflected over the bar while Tim Cahill did well to deny Stiliyan Petrov with a last-ditch tackle.
Slowly Everton began to find their feet and, in the 25th minute, Cahill's powerful shot flew straight at Brad Friedel.
Just 60 seconds later the veteran American keeper was forced to make a more routine stop from Marouane Fellaini's header.
Everton's revival was rewarded with a goal on the half-hour mark. After Carlos Cuellar had conceded a soft free-kick, Arteta rifled the ball in, Leon Osman got a flick to it and Lescott was on hand to knock it home from close range with his knee.
Everton were now on top and, three minutes before the break, Cuellar was forced to head Fellaini's effort off the line.
Moyes' side started the second half as they had ended the first - on top - and, in the 47th minute, Fellaini's close-range header came back off the crossbar via Friedel's fingertips.
However, Everton were rocked in the 54th minute when Jagielka's awful back-pass fell straight to Ashley Young who had little trouble chipping the ball over Howard and into the back of the net.
For all Everton's late pressure, it appeared that Villa would win the game 2-1 until Lescott's spectacular strike brought the sides level.
Amazingly Villa were not finished as Ashley Young used his pace to beat Lescott and Joseph Yobo before firing past Howard to pile more Goodison misery on hapless Everton.