Celtic opened the scoring after just five minutes, when Callum McGregor poked the ball into an empty net after some persistent play from Jo Inge Berget on the left-flank.
Less than ten minutes later, the Slovenian side levelled the scores through a tidy Damjan Bohar finish, who burst the offside trap to collect a superb Zeljko Filipovic through-ball and place it under a sprawling Craig Gordon.
Hoops skipper Charlie Mulgrew and defender Virgil van Dijk came close before the interval as the Scottish champions showed a marked improvement on their Legia Warsaw performances, where they were beaten 6-1 on aggregate in the third qualifier before the Polish side's disqualification.
Maribor came out with more purpose at the start of the second half in the Ldjuski vrt stadium but it was Celtic's Stefan Johansen who passed up a great chance to win the game when he stumbled in the area after bursting past a couple of challenges.
The Bhoys had Gordon to thank, however, with the former Sunderland stopper keeping out numerous chances - notably Ales Mertelj's stinging 30-yard drive in the latter stages of the game.
Deila's men will now be in pole position for group stage qualification, with the Scottish side set to entertain their Slovenian counterparts at Celtic Park in the return leg next Tuesday.
Deila dropped last season's top scorer Kris Commons to the bench, replaced by Beram Kayal, while right-back Mikael Lustig came in for Efe Ambrose, who joined up with the squad late after it was originally believed he was suspended for the game.
In the fifth minute, midfielder Johansen's raking pass picked out Jo Inge Berget on the left and when he got his shot in at the second attempt, Maribor goalkeeper Jasmin Handanovic could only palm the ball to 21-year-old McGregor - who slammed in from eight yards for his third European away goal this season.
Maribor looked shaky in the following minutes but their lifeline came when Zeljko Filipovic split the Hoops defence wide, allowing Bohar to ignore a foul on Maribor skipper Tavares and go through and slip the ball past Parkhead keeper Craig Gordon.
Mulgrew almost restored the Hoops' lead in the 21st minute when he got his head on to a Johansen corner - but Filipovic cleared to safety.
Moments later, from another Johansen corner, Lustig headed over at the back post, while Gordon made a decent save from Petar Stojanovic's angled drive at the other end.
Just after the half-hour mark, Parkhead stopper van Dijk beat Handanovic with a header from an Anthony Stokes corner, only to see the ball rebound off Johansen - who was on the line and consequently flagged offside.
The home side could have gone in front four minutes from the break when Gordon parried a long-distance drive from Agim Ibraimi into the air - but Dare Vrsic volleyed high over the bar.
Stokes had the ball in the Maribor net early in the second half but even he knew it was from an offside position, before Tavares fired a yard over the bar from the edge of the box after being set up by Vrsic, the first sign that the home side had upped the tempo.
Celtic looked happy to contain the Slovenian champions but Gordon had to make a fine save in the 59th minute from Ibraimi's powerful header, after Hoops left-back Emilio Izaguirre had needlessly conceded possession.
There was a Hoops penalty claim soon afterwards when Stokes' header from a McGregor cross appeared to strike the hand of Maribor stopper Aleksander Rajcevic - but Czech referee Pavel Kralovec was unimpressed.
As the home side loosened their grip on the game, Stokes twice shot wide, the second time just inches past the post - but Johansen should have scored when he was sent clean through on goal by Kayal but he lost control of the ball, allowing Handanovic to gather.
As Celtic finished strongly, Handanovic made a decent save from van Dijk after Kayal had gone down in the box under a challenge by Mitja Viler, another penalty claim ignored.
With seven minutes remaining Gordon had another impressive save from Ibraimi before surviving a late burst of pressure from the home side.