Blackburn Rovers secured their place in next season's Uefa Cup after beating a much-changed Chelsea to finish sixth.
Savage was booked for over celebrating Reid's goal
Both sides had hit the woodwork - Craig Bellamy for Blackburn and Eidur Gudjohnsen for Chelsea - before Steven Reid headed in from a free-kick.
In the second half champions Chelsea looked more dangerous, with Gudjohnsen going close with a volley, as did Ricardo Carvalho after a corner.
Substitute Joe Cole was also denied by Brad Friedel's save late on.
Chelsea's team included Frank Lampard, who had wanted to play one game before the World Cup, despite manager Jose Mourinho's offer to rest him.
Lampard's commitment was commendable but in a radically-reshaped side - there were eight changes from the team that wrapped up the title on Saturday.
After the interval Mourinho withdrew Asier del Horno, replacing the Spanish defender with former Blackburn winger Damien Duff, in an attempt to bring some pep to the Chelsea attack.
Cole, another England player Mourinho had suggested would not be used in Chelsea's final two league games, also replaced the ineffective Shaun Wright-Phillips midway through the second half.
Duff's introduction provided Chelsea with more width, but despite plenty of possession the visitors were unable to come up with a goal.
Blackburn went into the game with the incentive of knowing they could secure a European berth before the final game of the season and had gone busily about their business from the start.
Bellamy led by example with some intelligent running and the Welshman was well supported by Morten Gamst Pedersen.
Bellamy was unlucky not to score with an inventive effort, flicking the ball onto the woodwork after Michael Gray had crossed.
Bellamy's effort rebounded to David Bentley, who with the goal at his mercy, headed wide.
Just before Bellamy struck the bar, Geremi had gone close for Chelsea with a low shot that went narrowly wide.
Geremi then swung in a dangerous cross from the right and Gudjohnsen lunged to get a toe on the ball, his effort richocheting off the upright.
Having survived that scare Blackburn went ahead two minutes before the interval.
After Geremi was penalised and booked for handball, Savage's free-kick was glanced home by Reid.
Unaware Reid had got the final touch, Savage rushed to the Blackburn fans to celebrate and was booked by referee Steve Bennett for his efforts.
Chelsea began the second half in a more adventurous mood and Gudjohnsen's volley drew a fine reaction save from Friedel.
Chelsea might also have had a penalty after Rovers substitute Aaron Mokoena tugged Hernan Crespo to the ground in the box, but the offence went unpunished as Rovers held on.
Frank Lampard and Wright-Phillips earlier had had penalty appeals turned down.
Blackburn boss Mark Hughes:
"We're absolutely delighted. It's been a huge effort by a lot of people. Sometimes you don't get what you deserve in life - but we have.
"We need to keep moving forward. We came from a point last year where we were rock bottom of the Premier League and were in danger of going down.
"This year we have built on the qualities we had to show last season."
Chelsea manager Jose Mourinho:
"Two penalties, that is 100% - for sure. Maybe a third.
"I've had a difficult ride in the season in relation to this. We have survived, but this was too much.
"Blackburn deserve to be in Europe - they have done fantastically well to be here. But they don't deserve to win like this."
Blackburn: Friedel, Reid (Mokoena 72), Khizanishvili, Neill, Gray, Bentley, Savage, Tugay (Peter 85), Pedersen, Sinama Pongolle (Kuqi 85), Bellamy.
Subs Not Used: Enckelman, Dickov.
Booked: Savage, Gray.
Goals: Reid 43.
Chelsea: Cudicini, Geremi (Carlton Cole 77), Ricardo Carvalho, Gallas, Del Horno (Duff 45), Maniche, Diarra, Lampard, Wright-Phillips (Joe Cole 64), Crespo, Gudjohnsen.
Subs Not Used: Pidgeley, Johnson.
Booked: Geremi, Duff.
Ref: S Bennett (Kent).