Bidding to become the first team to retain the Champions League trophy, Chelsea went 2-0 up through a brace from Brazilian starlet Oscar at Stamford Bridge but had to make do with a point after Juve fought back to draw their Group E opener.
Oscar (21) was making his first start since a £25-million move from Internacional and he announced his arrival in style.
In the 31st minute he beat Gianluigi Buffon with a deflected effort from 25 yards and two minutes later he doubled Chelsea's lead with a stunning goal, sweeping a glorious shot into the top-right corner from outside the box.
Arturo Vidal pulled a goal back before half-time with a low shot into the bottom-right corner but Juve's equaliser did not arrive until the 80th minute, when Claudio Marchisio released substitute Fabio Quagliarella to beat Petr Cech in the Chelsea goal.
"We found ourselves in a great position to win the game and we only got a draw, but we knew we were playing a quality side," Chelsea coach Roberto Di Matteo told Sky Sports. "We feel a little deflated because we were ahead but it's going to be a tough group."
The result in west London meant that Shakhtar Donetsk finished the evening top of Group E following a 2-0 win at home to Danish competition debutants Nordsjaelland.
United took a seventh-minute lead against Galatasaray at Old Trafford when Michael Carrick swapped passes with first Robin van Persie and then Shinji Kagawa before skipping past goalkeeper Fernando Muslera and sliding the ball home.
Foot of the pedal
The English giants – eliminated at the group stage last season – almost paid the price for taking their foot off the pedal, however, with Nordin Amrabat hitting the bar and Hamit Altintop clipping the post for the visitors before half-time.
Nani squandered an opportunity to give United breathing space when his 53rd-minute penalty was saved by Muslera, while the home side were indebted to a fine double save from David de Gea that saw him deny Burak Yilmaz and Emre Colak.
"They were very confident in their possession, and missing the chances we did in the second half kept us on the edge," said United coach Sir Alex Ferguson.
"Three points in the opening game are very important, especially when you look at what happened to us last season."
In the other Group H game, United's next opponents, Romanians CFR Cluj, won 2-0 at Portuguese side Braga courtesy of a first-half brace from former Braga forward Rafael Bastos.
Barca suffered an early blow at Camp Nou when Gerard Pique had to go off injured, but the Catalans – champions in 2006, 2009 and 2011 – went ahead when Cristian Tello curled a low shot home from the cusp of the 18-yard area.
A Dani Alves own goal gifted Spartak a 29th-minute equaliser and the Russians sensationally took the lead just before the hour when Romulo beat Victor Valdes from Aiden McGeady's pass.
On the verge
The Russians stood on the verge of a famous upset, only for Messi to equalise from Tello's lay-off in the 71st minute before heading in Alexis Sanchez's cross 10 minutes from time to hand Tito Vilanova victory in his first Champions League game as coach.
"The match became complicated for us and it wasn't easy," Messi told Canal+.
"My goals aren't important. What's important is that the team keeps this dynamic."
In the night's other Group G fixture, Celtic were held to a 0-0 draw at home to Benfica on the Scottish side's return to the group phase after a four-year absence.
Bayern Munich assumed control of Group F, winning 2-1 at home to Valencia – who lost on penalties to Bayern in the 2001 final – through goals from Bastian Schweinsteiger and Toni Kroos.
There was a disastrous start to the group phase for 2011 French champions Lille, beaten 3-1 at home to Belarusian outfit BATE Borisov, who had never previously won a game in the competition. – Sapa-AFP