Atletico looked to have secured a trademark 1-0 win in unspectacular circumstances when Alvaro Morata scored 20 minutes from time, but Perez levelled in style, making it five league outings in a row with a goal for the West Ham flop.
During a dismal first half, Atletico looked utterly devoid of intent, while Alaves often appeared clueless in possession, only looking even moderately threatening from distance or with hopeful long balls into the box.
Proceedings livened up a little after the break and substitute Morata had seemingly put his team on course for a victory that would have put Atletico top for at least a few hours, but Perez struck with seven minutes to go to salvage a commendable point for the hosts.
Arguably the only form of entertainment before the break came in the tussle between Victor Laguardia and Diego Costa, who clashed multiple times.
One such occasion left Costa - surprisingly restored to the starting XI in place of the in-form Morata - adamant he should have been given a penalty after being thrown to the floor off the ball, though replays suggested the Spain striker made the most of the contact.
Atletico finally displayed some creativity early in the second half, but Renan Lodi inexplicably shot well wide from close range after latching on to Hector Herrera's lofted throughball.
Morata - who replaced Herrera - made no such mistake when released by Angel Correa in the 70th minute, racing clear and placing through Fernando Pacheco's legs to seemingly seal all three points.
However, the visitors spurned further opportunities to extend their advantage before Perez continued his incredible run of form with a gorgeous equaliser, curling a left-footed strike into the top-left corner from just outside the box.
What does it mean? Lack of creativity costs Simeone
While Costa was undoubtedly poor, it is fair to say his supporting cast did not provide particularly great service to him.
Only when the match began to open up a little did Atletico begin to create – had they been sharper earlier, Perez's late goal could have been a mere consolation and Simeone's men would have been sitting pretty at the summit.
Theatrical Costa looks a shadow of his former self
In years gone by, this was the sort of contest Costa relished, as the two formations had him pitted up against a centre-back out for a scrap in Laguardia. But the former Chelsea star looked more intent on throwing himself to the floor and offered precious little threat before being withdrawn coincidentally 18 seconds before Morata's goal.
Morata proves a point again
Simeone's decision to leave Morata on the bench in favour of Costa raised eyebrows pre-match. Given the marked contrast in their performances – the former looking extremely sharp off the bench – it is difficult to see how the coach can make that error again. The only sour note was the 27-year-old missing a good chance at 1-0 when clean through.
Key Opta facts
- Lucas Perez has become the second Alaves player to score in five consecutive games in LaLiga history (five goals), after Wilson in February 1955 (six goals)
- Morata has scored in three consecutive matches for Atletico for the first time
- Atleti are unbeaten in their seven LaLiga matches against Alaves under Simeone (W4 D3)
- Correa has provided three assists in his last two LaLiga games - that is as many as he had in his previous 46 in the competition
Atletico face another away match on Saturday as they travel to Sevilla, while Alaves go to Pamplona the following day for a clash with Osasuna.