Elber hit two goals in each half and set up another, to help Bayern wake up in style a week after opening their Bundesliga campaign with a dismal goalless draw with Borussia Moenchengladbach.
German midfielder Michael Ballack scored his first goal since moving from Bayer Leverkusen, while Peru striker Claudio Pizarro also played his part in a victory that lifted Bayern up to fourth, two points off the pace.
Promoted Bielefeld scored twice through Polish striker Artur Wichniarek and Senegal forward Mamadou Diabang but never stood a chance.
Defending champions Dortmund fought back from a goal down to a 1-1 draw away to last season's runners-up Leverkusen on Saturday, a result that left both teams without a win after two games and four points behind surprise leaders VfL Bochum.
Leverkusen looked in control for most of the match, opening the scoring with an angled effort from Argentine wing-back Diego Placente in the 23rd minute, but a headed goal from Dortmund's Czech striker Jan Koller on 61 minutes denied them victory.
Promoted Bochum, who had started with a 3-1 win at Nuremberg, impressed again when a hat-trick by striker Thomas Christiansen inspiring them to a 5-0 thrashing of Energie Cottbus.
Bochum, who have been relegated four times over the past 10 years but always managed to make it back to the top flight, lead on goal difference from Hansa Rostock and Schalke 04 the only other teams with a maximum six points.
Northerners Rostock sent Nuremberg to the bottom of the table with a 2-0 victory while Schalke pulled off a spectacular 3-1 win at Kaiserslautern.
In France, the in-form En Avant Guingamp clinched a convincing 3-0 home victory over Stade Rennes to lead the French first division on Saturday.
Elsewhere, an incident-packed day produced no less than eight red cards, one of which was for Bastia striker Florian Maurice his second in three games since the start of the season.
Bastia's 1-0 victory over Monaco was also marred by a brawl which stopped play for five minutes and a further red card for Monaco's Adbelrahmame Mazar.
Nantes were another to suffer red cards, as they slipped to a 4-2 defeat by Sochaux ending the match with nine men.
Promoted Le Havre will have to learn how to behave in the elite as they too finished with only nine players as William Mocquet and Pierre Ducrocq were sent off for second bookable offences. The Normandy side still managed to hold on for a 1-1 draw at Strasbourg.
Guingamp had only two players booked as they crushed Stade Rennes with striker Florent Malouda netting twice in the 33rd and 86th minutes. Christophe Leroux was also on target for the Brittany side.
Guingamp are on seven points, level with Girondins Bordeaux who snatched a 1-0 success away at Montpellier despite playing the second half with only 10 men after David Jemmali was sent off. Alexei Smertin scored the game's only goal after 20 minutes.
Surprise package Nice, promoted last season, upset Lille 3-0 to move up to third position while a late goal from Djibrill Cisse handed AJ Auxerre a 2-1 win at troubled Troyes. French champions Olympique Lyon drew 1-1 at Olympique Marseille on Friday.
In Holland, PSV Eindhoven, under new coach Guus Hiddink, opened their Dutch league campaign with a 2-0 win over newcomers Excelsior Rotterdam on Saturday.
The young and nervous Excelsior side whose average age is around 21 were no match for PSV, runners up last season, and the game was virtually decided after only 17 minutes.
Captain Mark van Bommel took advantage of a defensive error to put PSV ahead after three minutes and 14 minutes later Arnold Bruggink curled a free kick over the wall and out of reach of goalkeeper Zbigniew Malkowski for the second.
PSV should have scored more goals, but lack of concentration and sloppy passing on a hot summer's evening saved Excelsior from total humiliation.
PSV's unimpressive game a week after losing the Dutch Super Cup match 3-1 to Ajax gives new coach Guus Hiddink who steered South Korea to fourth place in the World Cup, work to do on his attack.
In other matches on Saturday, AZ Alkmaar and Heerenveen split the points in a 3-3 draw and NAC Breda and Roda JC Kerkrade drew 2-2.
Willem II Tilburg sank Twente Enschede 2-0 with second half goals by Tarik Sektioui and Dmitri Shoukov.
Two promoted sides, RBC Roosendaal and Zwolle, kicked off the first division championship with a 1-1 draw on Friday.