Odobert sent Burnley ahead in the closing stages of the first period, but Adingra levelled the scores in the 77th minute, with the spoils being shared at the Amex.
Brighton sit eighth in the Premier League table, one point behind fifth-placed Tottenham Hotspur, while Burnley occupy 19th, level on points with basement side Sheffield United.
Burnley had the first opportunity of the match in the 10th minute, with Johann Berg Gudmundsson finding space just inside the penalty box, but his effort flashed past the post.
Gudmundsson then turned provider in the 19th minute, with the experienced attacker finding Zeki Amdouni on the edge of the box, but his effort was smartly saved from Brighton goalkeeper Bart Verbruggen.
Brighton were working their way into the match, with Adam Lallana missing the target in the 22nd minute, before Burnley defender Charlie Taylor managed to clear a dangerous cross ahead of Adingra, who looked on course to tap home from close range.
Burnley goalkeeper James Trafford was forced to make a smart save to deny Pascal Gross in the 28th minute, with the Brighton midfielder looking to find the bottom corner.
Trafford made another excellent stop to deny James Milner, before a header from Jan Paul van Hecke flashed just wide of the target in the 35th minute, with the two teams remaining level.
Burnley managed to make the breakthrough in the latter stages of the first period, though, with Odobert finding the back of the net with an excellent finish from distance after creating the space and releasing a strike.
The visitors had a chance to double their lead early in the second period through Jay Rodriguez, who had been set up by Odobert, but Verbruggen was on hand to make a smart save.
Brighton came close to a leveller in the 63rd minute when Kaoru Mitoma found space in a wide area before looking to pick out the top corner, but his effort was just too high.
Evan Ferguson then headed wide from 10 yards out, with Brighton staring to turn the screw in search of an equaliser against Kompany's side.
The Seagulls managed to level the scores in the 77th minute, with Adingra heading a cross from Gross into the back of the net to register the equaliser that was deserved following a period of pressure.
Brighton were firmly on the front foot in the period that followed, with excellent defending from Dara O'Shea cleaning a low cross from Mitoma, who was a constant threat in the second period.
Trafford needed to stand strong in the latter stages, with the Burnley goalkeeper denying Jack Hinshelwood and Mitoma to help his side claim a welcome point at the Amex.