Willian sent Chelsea ahead just past the hour before Messi levelled for the visitors in the 75th minute to leave the tie delicately poised ahead of the return match at Camp Nou next month.
It was Chelsea that had the first half-chance of the match in the fifth minute, but Hazard's powerful strike from outside the Barcelona box was always rising over the crossbar. Antonio Rudiger then headed just wide of the post in the ninth minute as the home side had a couple of positive moments early on.
Barcelona's first serious opportunity arrived in the 15th minute when Messi found the head of Paulinho with a brilliant cross, but the Brazilian could only nod wide of the post.
The away side absolutely dominated the ball in the opening 30 minutes of action, but clear chances were few and far between as Chelsea's outfield players did an impressive job of protecting Thibaut Courtois.
Marcos Alonso curled a free kick into the arms of Barcelona goalkeeper Marc-Andre ter Stegen in the 31st minute as Chelsea registered their first attempt on target, but their best chances of the first period were still to come.
Indeed, Hazard worked space down the left in the 33rd minute before feeding Willian, who went past Sergio Busquets before curling a wonderful effort against the post. Willian then hit the other post eight minutes later as the Brazilian found space once again in a dangerous area.
Chelsea had another clear opportunity in the latter stages of the first period when the ball fell for Hazard inside the Barcelona box, but the Belgian could only send his volley over the crossbar.
Barcelona dominated the possession in the first 15 minutes of the second period, but the Spanish outfit were still struggling to create chances as Chelsea's back five continued to defend well.
Willian hit both posts in the first period, but the Brazilian made the breakthrough in the 62nd minute when he again worked space on the edge of the Barcelona box before bending a wonderful effort into the bottom corner of the net.
The pattern of the match did not change as Barcelona continued to enjoy plenty of the ball, but they were finding it increasingly difficult to carve out opportunities at Stamford Bridge.
A mistake from Chelsea brought Barcelona their equaliser, however, as a loose pass from Andreas Christensen was gobbled up by Andres Iniesta, who fed Messi, and the Argentine made no mistake with a calm finish.
Morata came off the Chelsea bench in the 83rd minute as Conte looked for an advantage to take into the second leg in Spain next month.
Victor Moses had one deflected wide of the Barcelona post in the 86th minute, but Chelsea were made to settle for a draw as a costly mistake handed the Catalan giants what could prove to be a vital away goal.
CHELSEA (3-4-3): Courtois; Rudiger, Christensen, Azpilicueta; Moses, Kante, Fabregas (Drinkwater 84'), Alonso; Willian, Hazard, Pedro (Morata 83')
BARCELONA (4-4-2): Ter Stegen; Roberto, Pique, Umtiti, Alba; Busquets, Rakitic, Iniesta (Gomes 90'), Paulinho (Vidal 64'); Messi, Suarez