An 87th minute strike from Robert Lewandowski gave the German champions all three points, while big-spending PSG coasted to a 4-1 win over Dynamo Kiev in Paris.
Champions League round-up: Dortmund snatch late victory over Ajax as AC Milan struggles continue
Late drama was the order of the day in Group D following Real Madrid's last minute win over Manchester City, as Robert Lewandowski's 87th-minute goal gave Borussia Dortmund a 1-0 victory over Ajax.
Mats Hummels' missed second half penalty looked like being costly for Jurgen Klopp's side but Lewandowski struck three minutes from time in front of a crowd of more than 80,000 at the Westfalenstadion.
Paris St Germain made a convincing start to their Champions League Group A campaign with a 4-1 victory over Dynamo Kiev at the Parc des Princes.
Carlo Ancelotti's men never looked back after establishing a three-goal lead inside 32 minutes, through Zlatan Ibrahimovic's penalty, Thiago Silva's close range finish and Alex's right foot blast from Clement Chantome's outswinging corner.
Although Miguel Veloso pulled a goal back four minutes from time, Javier Pastore wrapped up the scoring in injury-time as PSG ended an eight-year absence from European football's top club competition in style.
Porto won the other Group A match 2-0 away to Dinamo Zagreb.
Steven Defour put the seal on the win in injury-time after Luiz Gonzalez put the visitors in front four minutes before the break.
The win keeps Vitor Pereira's side unbeaten in all competitions in a run stretching back to a League Cup defeat at Benfica in March.
Schalke opened with a victory over Olympiakos in a closely-fought game in Greece in the other game in Arsenal's Group B.
Benedikt Howedes headed the German side in front four minutes before half-time and, after Djamel Abdoun equalised, Klaas Jan Huntelaar grabbed what proved to be the winner.
The Holland striker then missed a penalty but his side nevertheless made a winning start.
Malaga began their first ever Champions League campaign in perfect fashion as two stunning goals from Isco guided them to a 3-0 win over Zenit St Petersburg at La Rosaleda.
The game was barely three minutes old when the Spanish Under-21 international bent a wonderful shot in off the inside of the post and the hosts were in dreamland when Javier Saviola slid home a second 10 minutes later.
Zenit then missed a host of opportunities to get back into the game with Roman Shirokov and Alexander Kerzhakov the main culprits and their miserable evening was rounded off when Isco thumped in a third from outside the area with 14 minutes remaining.
The other Group C match saw AC Milan's struggles continue in a 0-0 home draw against Anderlecht.