Sevilla, who this week were knocked out of the Europa League, twice took the lead through goals from Alvaro Negredo and Ivan Rakitic but Atletico hit back both times to deservedly claim a point.
Teenager Jorge Koke cancelled out Negredo's strike with his first league goal for the club, while Reyes, who came up through the youth ranks at Sevilla, netted Atletico's second equaliser with 14 minutes to go.
Both clubs will be pleased to have avoided defeat against one of their main rivals for a Europa League place, but they could find themselves even further adrift of the top six if Athletic Bilbao beat Valencia tomorrow.
Atletico started the better but, after Reyes, Sergio Aguero and Raul Garcia had all tested visiting goalkeeper Javi Varas, it was Sevilla who took the lead four minutes before half-time.
Mouhamadou Dabo's deep cross was chested down by Diego Perotti to the edge of the area, where former Real Madrid striker Negredo struck an unstoppable left-foot shot past David De Gea.
Atletico will have been bitterly disappointed to have gone behind after looking on top, but they almost drew level in first-half injury-time when Koke poked Diego Forlan's low cross goalward, and it needed an outstretched foot from Varas to deny the youngster.
Varas was left helpless by Koke two minutes into the second half however as the 19-year-old got on the end of another Forlan cross to cushion a header just inside the far post.
That was the perfect start to the second period for Atletico, but they were fortunate to still be on level terms soon after as Negredo missed a great chance for Sevilla before Luis Fabiano saw a powerful header saved by De Gea.
As the game swept from one end to another, Atletico went close themselves in the 58th minute but Varas denied Aguero before Koke's follow-up was also blocked.
Atletico once again looked to be gaining the upper hand, but they fell behind for a second time in the 65th minute as Sevilla scored a superbly-worked breakaway goal.
Aguero was muscled off the ball in the Sevilla area by Julien Escude, and Gary Medel and Fernando Navarro combined to set Perotti away down the left. Perotti then cut inside Tomas Ujfalusi before picking out Jesus Navas, who promptly laid the ball back to the edge of the box where Rakitic side-footed into the far corner of the net.
Sevilla again could not hold onto their lead, however, as Atletico drew level with 14 minutes to go when Reyes was put through by Antonio Lopez and he slotted a low shot past Varas.
That was the end of the scoring, but it was not the end of the chances and both sides came close to snatching all three points in the closing stages.
Aguero had three good attempts, the final of which was a magnificent opening with 10 minutes to go after Varas had picked up a back-pass from team-mate Sergio Sanchez.
Atletico midfielder Tiago quickly grabbed the ball off the bemused Sevilla custodian and immediately passed to Aguero, but with a goal looking a certainty, Varas redeemed himself by throwing himself at the Argentinian's shot and managing to deflect it for a corner.
That was a huge let-off for Sevilla, but Atletico were also left breathing a sigh of relief in the 87th minute when Negredo's superb 25-yard effort came back off the underside of the crossbar with De Gea beaten.