Fulham and Birmingham both won on their travels, the Cottagers 2-1 against Odense and the Championship side came from behind to defeat NK Maribor 2-1, whilst Tottenham and Stoke both tasted home success. Celtic took an early lead from the penalty spot at home but then were undone in the final minute as visitor's Udinese levelled with their own spot kick.
Andy Johnson scored a goal in each half to give Fulham their first Europa League group stage win over Odense.
The host's miserable evening was complete as Kalilou Traore saw red in injury time.
Birmingham had to come from behind after a goalkeeping error from Colin Doyle allowed Dalibor Volas to score.
But the Blues rallied after the break to score twice from Chris Doyle and Wade Elliott to give Birmingham confidence after a disappointing start to life in the Championship.
Ki Sung-yueng scored from the spot in the third minutes after striker Gary Hooper had been fouled by Joel Ekstrand.
But the Italians increased the tempo after the break and finally restored parity in the 86th minute when Hooper gave away a penalty.
Tottenham survived a scare falling behind 1-0 to Irish outfit Shamrock Rovers before recovering for a comfortable 3-1 home success.
Rovers took an unlikely lead after the break when Stephen Rice scored.
The Turkish side opened the scoring in the 14th minute through Ricardo Quaresm, only for Peter Crouch to restore parity within a minute.
Walters then netted from the spot with 12 minutes remaining.
After drawing their first game in Kiev the Potters are now top of Europa League Group E.