The Blues’ title credentials were shown up again with a goal in each half by Dean Windass and debutant Benito Carbone securing victory.
The points were shared at the Riverside Stadium where Middlesbrough hit back to draw 1-1 with Tottenham.
Spurs took the lead four minutes before the interval through Norwegian international Oyvind Leonhardsen only for Mark Summerbell to level scores with a firm header in the 59th minute.
Wilnis rocked Manchester United as the champions made a stumbling start against plucky Ipswich.
Portman Road, already at fever pitch to welcome Ipswich back to the Premiership, erupted when Wilnis applied a thumping right-foot finish to a sweeping move, started by a quick midfield exchange between Jim Magilton and Matt Holland.
Barthez kept United in the game with a string of fine before United pulled level in the 39th minute.
Wilnis collected a booking for fouling Philip Neville wide on the United left and Beckham stepped up to take the free-kick.
The United winger swung a dipping ball right-footed into crowded goalmouth and home keeper Richard Wright was deceived by Ronnie Wallwork’s dummy and the ball bounced into the far corner of the net.
Dwight Yorke, booked as he left the field at half-time, was narrowly off target with a long-range strike after the re-start and Wright had to be alert to save from Ole Gunnar Solskjaer.
Wright saved again from Solskjaer after a forward run by Mark Venus had left the hosts stretched at the back after mounting a speedy counter-attack while Barthez saved from Martin Reuser as both teams searched for the winner to no avail.
Bradford’s attempt to take on their cosmopolitan opponents at their own passing game paid off as Chris Hutchings’ side took the lead after 24 minutes.
Lee Sharpe found space down the left flank and swung over a cross which Windass nodded back across Ed de Goey and inside the far post.
And Carbone sealed the game 16 minutes from time with a wonderous strike after a lapse from Chelsea substitute Christian Panucci.
The Italian full-back sold colleague Emerson Thome short with a pass just inside his own half and Windass instigated a breakaway.
The widely travelled striker found Carbone on the right-hand edge of the area and De Goey failed to keep out a fierce dipping 20-yard shot at the foot of his left-hand post.
Middlesbrough and Tottenham preserved their unbeaten starts to the new season in a hard-fought draw at the Riverside Stadium.
Leonhardsen put the visitors ahead with a 39th-minute header after George Graham’s side survived an encouraging opening by the Teesiders Sergii Rebrov crossed from the right and Norwegian Leonardsen rose to head into the top corner.
But the home side equalised with 59 minutes gone when Curtis Fleming got free down the right and delivered an inviting cross for the unmarked Summerbell, who rose majestically to fire a bullet header into the net.