Brazilian midfielder Oscar marked his first start for Chelsea with two long-range goals, but Arturo Vidal pulled a goal back close to halftime and substitute Fabio Quagliarella equalized late on.
"We are disappointed and a little bit deflated to get in a great position to win the game and come away with only a draw," Chelsea manager Roberto di Matteo said. "We should have closed the game."
Oscar scored his first when he received a short pass from Eden Hazard and arrowed in a shot that took a looping deflection off Leonardo Bonucci and flew past goalkeeper Gianluigi Buffon.
His second goal less than two minutes later had no element of luck about it, beating Andrea Pirlo to the ball and executing a flick before curling a delightful effort into the top corner from just outside the area.
"It's great to see we have a player like him in our ranks," Di Matteo said.
Shakhtar Donetsk's comfortable 2-0 win against Danish debutant Nordsjaelland took the Ukrainian side top of Group E, with Armenian forward Henrik Mkhitaryan scoring in each half with excellent finishes.
In Group G, four-time champion Barcelona had Messi to thank as the Argentina forward took his season tally to 10 goals in seven games courtesy of a devastating 10-minute spell at Camp Nou.
"To talk about Messi is to talk about excellence in football," Spartak coach Unai Emery said. "It is very tough to stop him."
Cristian Tello opened the scoring for Barca in the 14th minute, before Dani Alves' own goal in the 29th and Romulo's strike in the 59th gave the visitors an unexpected advantage.
But Messi, who scored 14 goals in last season's campaign, tapped in Tello's pass in the 71st and headed in Alexis Sanchez's cross in the 80th to keep Barcelona undefeated in their last 17 home games in Europe.
Last April, Messi's failure to finish in the semifinal against Chelsea left him distraught and cost Barcelona a place in the final.
For a while, Spartak looked like giving him more cause for disappointment, but he dug deep to give Tito Vilanova an enjoyable European debut as Barcelona coach since taking over from Pep Guardiola.
"I think this game will do us good. It is easy to think that since we are Barcelona we will win with ease," said Vilanova, who helped Barcelona win the trophy in 2009 and '11 as Guardiola's assistant. "We were able to come back and win and that shows that the team still has the desire to win."
In the other group match, there was far less excitement as Celtic created little at Parkhead in a 0-0 draw with Benfica.
Manchester United made a good start following midfielder Michael Carrick's seventh-minute goal, but failed to impress in a 1-0 win against Galatasaray. Portugal winger Nani missed a second-half penalty as United struggled to get going at Old Trafford.
The Turkish side can consider itself unlucky, hitting the goal-frame three times, having two penalty appeals rejected and forcing a couple of good saves from David de Gea.
"I think we expected a tough game. Galatasaray have a lot of experience," United manager Alex Ferguson said. "There was a lack of concentration at times and it's unusual for us to be as frivolous with the ball."
Ferguson was keen to avoid a repeat of last year's embarrassment in which United fielded below-strength teams against reputedly weaker opposition and tumbled out at the group stage, but he still made six changes from Saturday's 4-0 win against Wigan.
Although Shinji Kagawa and Robin van Persie returned to boost the attack, it was Carrick who settled the early nerves. He was tripped by Fernando Muslera as he rounded the Galatasaray goalkeeper, but still slotted the ball into the net while trying to regain his feet.
Ferguson has been rotating his 'keepers this season, but can thank De Gea for a superb double save late on from Burak Yilmaz and Emre Colak.
Romanian side CFR Cluj moved top of Group H after winning 2-0 at Braga from two first-half goals by Brazilian midfielder Rafael Bastos, who was once rejected by the Portuguese side.
Bayern Munich, last season's runner-up, missed a late penalty in its 2-1 win against Valencia in Group F.
Bayern midfielder Bastian Schweinsteiger opened the scoring in the 38th minute and teammate Toni Kroos' powerful long-range strike gave the home side some breathing space.
"Of course it's always dangerous when it's 1-0. The second goal was very important for us," Schweinsteiger said.
Valencia substitute Nelson Valdez set up some late drama with an injury-time header, and there was still time for Bayern forward Mario Mandzukic to miss a penalty on his Champions League debut.
Belarusian champion BATE Borisov won 3-1 at Lille in the other Group F match, stunning Lille through a slick display of counterattacking with former Arsenal midfielder Aleksandr Hleb pulling the strings.
Midfielder Aleksandr Volodko, striker Vitali Rodionov and midfielder Edgar Olekhnovich scored in the first half. Center back Aurelien Chedjou pulled a goal back on the hour mark for the dispirited home side, which failed to progress from the group stage last season.