The English Premier League side were unable to add to Theo Walcott's fourth minute goal, leaving them with a tricky task in next week's return in Italy.
Benfica, the only former European champions in action, drew 2-2 at Twente Enschede while Olympique Lyon, hoping to qualify for the 12th time in succession, conceded an early goal before beating Russian visitors Rubin Kazan 3-1.
Viktoria Plzen have a great chance of qualifying for the first time after a 3-1 win at FC Copenhagen while Austrian title holders Sturm Graz held Belarus champions BATE Borisov 1-1 away.
Arsenal, with manager Arsene Wenger banished from the touchline and captain Robin van Persie and Samir Nasri also suspended, got the perfect start against opponents playing their first competitive game of the season.
Bacary Sagna released Aaron Ramsey on the right and his pinpoint cross was met first time by Walcott.
Wenger gave instructions by mobile phone from the directors' box but was ordered to stop at half-time by UEFA officials, Arsenal said.
Udinese, who sold key players Alexis Sanchez and Gokhan Inler in the close season, could have collapsed but instead hauled themselves back into the match and in the end deserved a draw.
Captain Antonio Di Natale went close twice, hitting the bar with a free-kick in the first half and also clipping a post with a shot in the second.
Colombian Pablo Armero should have scored after a 60-metre run on goal ended with him firing straight at Wojciech Szczesny.
Arsenal could also have had a bigger cushion for the second leg but Samir Handanovic made an outstanding save low to his left to deny Walcott from close range in the dying minutes.
"The most important thing was the clean sheet," Walcott said on Sky Sports. "They are a very strong team and we will find it difficult over there but I believe, if we play the same kind of football, we can get a result."
Udinese coach Francesco Guidolin told reporters: "I am disappointed we did not score and conceded a goal.
"But I think the tie is still very open and is there for the winning next week. However, I am disappointed that we had seven eight or nine chances and did not score."
RUIZ STRIKES
Benfica, who despite winning the old European Cup twice and finishing runners-up five times have made only a modest six Champions League appearances, were denied an away win when Costa Rican Bryan Ruiz scored an 80th minute goal for Twente.
The Dutch side had taken a sixth-minute lead through Luuk de Jong before Oscar Cardozo, with a curling shot, and Nolito put the Portuguese 2-1 ahead at half-time.
Rubin Kazan took a third-minute lead at Lyon through Vladimir Dyadyun but the French replied within seven minutes through Bafetimbi Gomis and a Solomon Kverkvelia own goal put them ahead five minutes before the break.
Jimmy Brian
