Ten-time champion Real Madrid trailed early against Bulgarian newcomer Ludogorets Razgrad, which was only formed as a club 13 years ago — in the same season the Spanish giant won its ninth title.
After Ludogorets' stunning sixth-minute opener, Cristiano Ronaldo had a penalty saved and then scored with a second spot-kick in the 24th, before Madrid got substitute Karim Benzema's 77th-minute goal to force a 2-1 win.
"We played as equal to Real and I am delighted with our performance," Ludogorets coach Georgi Dermendzhiev said.
Five-time winner Liverpool lost 1-0 at Basel, which specializes in beating English opponents, and trails Group B leader Madrid by three points before hosting the title holder next.
"I thought we were too soft all over the pitch," Liverpool captain Steven Gerrard said, adding his team "didn't deserve anything" against Switzerland's champion.
Borussia Dortmund won 3-0 at Anderlecht, and leads Group D by three points from Arsenal, which beat Galatasaray 4-1 despite goalkeeper Wojciech Szczesny being sent off with 30 minutes left.
Atletico Madrid, runner-up last season, beat Juventus 1-0 through Arda Turan's 75th-minute goal, and Malmo's 2-0 victory against Olympiakos left Group A with all four teams on three points.
Zenit St. Petersburg and Monaco drew 0-0 to share the Group C lead with four points, while Bayer Leverkusen beat Benfica 3-1.
Ludogorets' fans have to travel 335 kilometres (210 miles) to watch their team play at the national stadium in Sofia because the club's 6,000-seat arena does not meet Champions League standards.
But the supporters soon had reason to celebrate after Brazilian midfielder Marcelinho headed in a cross flicked on to the far post.
Ronaldo had a 10th-minute penalty saved by goalkeeper Vladislav Stoyanov, but soon got a second chance to convert and close on Raul Gonzalez's record of 71 Champions League goals in the group stage and knockout rounds. Ronaldo has scored 69 goals — not including one in a qualifying-round match — and leads Lionel Messi by one in what promises to be a season-long duel.
Veteran Basel forward Marco Streller has a more modest record yet his 52nd-minute strike against Liverpool followed his winning goal at Chelsea last season. Basel also eliminated Manchester United three seasons ago.
"It was a perfect evening," said Basel coach Paulo Sousa, hailing a "win against an historic club like Liverpool."
Arsenal is yet to be grouped with Basel, and struck a blow for the Premier League by extending Galatasaray's winless streak in England to 10 matches.
On Arsene Wenger's 18th anniversary as Arsenal manager, his two big off-season signings got goals to ease any anxiety caused by a group-opening 2-0 loss at Dortmund.
Danny Welbeck, signed from Manchester United at the transfer deadline, scored in the 22nd, 30th and 52nd minutes. Former Barcelona winger Alexis Sanchez had made it 3-0 in the 41st.
Galatasaray's fans rather than players offered fireworks in the first half. Play was briefly stopped before Sanchez's goal to allow red smoke to clear after numerous flares were lit by the Turkish club's fans.
Only when Szczesny was sent off, for bringing down Burak Yilmaz, did Galatasaray begin to threaten. Yilmaz sent the resulting 63rd-minute penalty past substitute goalkeeper David Ospina, signed after impressing for Colombia at the World Cup.
Dortmund again eased to victory, though had only Ciro Immobile's third-minute goal to show for dominating the first half in Brussels. The Italy forward finished a pass from Japan playmaker Shinji Kagawa who, like Welbeck, was unwanted at Old Trafford by new manager Louis van Gaal.
Dortmund added two second-half goals from Adrian Ramos, another recruit from Colombia's World Cup squad.
In a match of few chances in Madrid, Atletico scored a rare goal past Juventus 'keeper Gianluigi Buffon when Turkey forward Turan connected at the far post.
Malmo was a European Cup finalist in 1979, but is playing in its first Champions League and made a winning home debut against Olympiakos when captain Markus Rosenberg scored late in each half.
"We're strong at home and we're not afraid," Rosenberg said.
Zenit St. Petersburg and Monaco played out a drab 0-0 draw in Russia, leaving Leverkusen to do the entertaining in Group C.
The German side, which lost in Monaco last month, closed the gap with a comfortable win against visiting Benfica. Veteran forward Stefan Kiessling scored in the 25th , after Brazil 'keeper Julio Cesar fumbled a shot by Son Heung-min, and the South Korea forward extended the lead nine minutes later.
After Benfica's Eduardo Salvio pulled a goal back in the 62nd, Leverkusen restored its two-goal lead two minutes later when Hakan Calhanoglu converted a penalty.Suggest a correction