There were good wins for Barcelona and Bayern Munich as Manchester City's game was postponed after a torrential downpour in Manchester.
Arsenal travelled to the Parc des Princes for their clash with Ligue 1 champions Paris Saint-Germain, where they appeared to be undone by a first-minute Edison Cavani goal.
The Gunners worked hard to break down their opponents and had goalkeeper David Ospina to thank as he stopped PSG from recording a more dominant result, before Alexis Sanchez popped up on 77 minutes to smash the ball past Alphonse Areola after the keeper had made a good close-range save to deny Alex Iwobi.
Both teams finished with 10 men as Oliver Giroud and Marco Verratti were dismissed after picking up a second yellow card each for a pointless off-the-ball incident as the Gunners striker threw an arm across the PSG winger as he tried to evade his challenge.
FC Basel, meanwhile, succumbed to a 45th-minute Jonathan Cafu goal in their clash with Ludogorets before finding an 80th-minute equaliser with a fine volley from the boot of Renato Steffen with just 10 minutes of the match remaining.
Elsewhere, two goals from Arkadiusz Milik saw Napoli overcome Dynamo Kiev 2-1. Milik opened the scoring on 36 minutes before adding a second two minutes into first half stoppage-time to cancel out Denis Garmash's 26th-minute opener.
Benfica looked to be heading for a narrow win at home to Besiktas with Franco Cervi netting in the 12th minute – with an easy finish from close range after goalkeeper Tolga Zengin parried an Eduardo Salvio shot into his path – but Talisca ruined the party for the Portuguese side as he scored deep into injury time.
Lionel Messi and Co. ran riot as Barcelona thumped Celtic 7-0 with the Argentine frontman getting off the mark after just three minutes before a 24th-penalty miss from Moussa Dembele. That proved to be costly as Barca upped the tempo and kept hammering home the goals.
Messi added his second on 27 minutes before Neymar got in on the action with 50 minutes played. Andres Iniesta then netted nine minutes later while Messi completed his hat-trick on the hour mark.
Luis Suarez added to the scoreline with an audacious volley after controlling the ball on his chest before spinning and smashing home. The Uruguayan rounded it all off with 88 minutes played with a simple swept finish when unmarked in the box, from a square Messi pass.
Bayern Munich where too strong for Rostov as they recorded a 5-0 win at the Allianz Arena where Robert Lewandowski got his side underway with a 28th-minute penalty.
Thomas Muller added a second in stoppage-time at the end of the first half before Joshua Kimmich scored in the 53rd minute and again with 60 minutes played. Juan Bernat then completed the rout with his 90th-minute effort.
And in the final match of the first round of group games, Saul scored the only goal of the tie on 43 minutes as Atletico Madrid defeated PSV 1-0 at the Philips Stadion.
PSG 1-1 Arsenal
Basel 1-1 Ludogorets
Dynamo Kiev 1-2 Napoli
Benfica 1-1 Besiktas
Barcelona 7-0 Celtic
Manchester City v Borussia Monchengladbach (postponed)
Bayern Munich 5-0 Rostov
PSV 0-1 Atletico Madrid