Arsenal, VfB Stuttgart and Olympiakos Piraeus all scored 2-0 away wins in their final Champions League playoff matches on Tuesday to take a big stride towards the group stage before next week's second legs.
Arsenal became the first English club to win a European tie at Celtic since Nottingham Forest in 1983 with a deflected William Gallas effort and an own goal from Gary Caldwell giving them victory.
Stuttgart won at Politechnica Timisoara in Romania with two goals in three minutes from Timo Gebhart, who scored with a penalty, and former Arsenal midfielder Aleksandr Hleb.
Goalkeeper Jens Lehmann, who played for Arsenal before joining Stuttgart, announced before the match that he was retiring at the end of the season.
Olympiakos also scored a good away win and dented Sheriff Tiraspol's hopes of becoming the first Moldovan team to qualify for the group stages with Dudu and substitute Kostas Mitroglou on target to give the Greek double winners their 2-0 victory.
In the two other matches, FC Copenhagen beat APOEL Nicosia 1-0 with a 51st-minute goal from Zdenek Pospech while Portugal's Sporting had their first goalscorer, Montenegro striker Simon Vukcevic, sent off for celebrating his strike during their 2-2 home draw with Fiorentina.
He earned a second yellow card for taking his shirt off after scoring Sporting's 58th-minute equaliser and now misses next week's evenly-balanced second leg in Italy.
Juan Manuel Vargas put Fiorentina ahead after six minutes and after Vukcevic equalised, Miguel Veloso put Sporting 2-1 up with a thunderous 25-metre shot before Alberto Gilardino equalised for the visitors 11 minutes from time.
Arsenal took the lead at Celtic when Cesc Fabregas's 43rd-minute free-kick was deflected past Celtic keeper Artur Boruc after bouncing off the back of Gallas who appeared to be crouching to get out of the way.
The goal may have had an element of luck but Arsenal had enjoyed the better of the play and confirmed their superiority when Caldwell, sent off playing for Scotland last week, endured more bad luck when he deflected a cross into his own net after 71 minutes.
"I felt we had more control of the ball before we scored and they did not trouble us defensively but it was a real cup game today and we needed some mental strength to get through that," Arsenal manager Arsene Wenger told Sky Sports News.
Celtic coach Tony Mowbray added: "I think Gallas was trying to get out of the way for their first goal but it knocked the stuffing out of what we were trying to do. Arsenal gave us a lesson with and without the ball and a lot of our players are disappointed with their own performances."
The remaining playoff first legs are being played on Wednesday.