Alvarez bags penalty double as Atletico beat Valladolid
Coach Diego Simeone's son Giuliano and Alexander Sorloth were also on target as the Rojiblancos moved back within seven points of league leaders Barcelona.
Atletico, third, also trail rivals Real Madrid by three points and still hold onto slender hopes of winning the title.
"We can't do anything more than keep moving forward, thinking about what's to come," Alvarez told DAZN.
Valladolid, without a win in 12 matches, gave Atletico a more uncomfortable evening than they would have imagined or hoped for.
The visitors took the lead through a Mamadou Sylla penalty after Clement Lenglet handled a corner kick as he went to head it clear.
However the hosts were quickly level with a penalty of their own after Javi Sanchez fouled Simeone.
Alvarez, who controversially had a penalty ruled out for supposedly touching the ball twice against Real Madrid as his team were knocked out of the Champions League, stroked it down the middle.
Two minutes later Simeone sent Atletico ahead, surging inside from the right and blasting home after Pablo Barrios sent a ball over the top for him.
Atletico were dominant but could not add to their lead and Lenglet allowed Valladolid back in with a foul outside his area.
Sanchez took the free-kick and it deflected off Conor Gallagher's shoulder in the wall past the helpless Jan Oblak.
Atletico were handed a chance to reclaim the lead when Marcos Llorente turned into the box and was clumsily brought down by Henrique.
Alvarez tucked home the game's third spot kick, sending Andre Ferreira the wrong way.
It was his 14th league goal of the season and 26th across all competitions after joining from Manchester City last summer.
Sorloth wrapped up Atletico's win with another goal off the bench, as has been his penchant this season, mopping up after Ferreira saved an Alvarez effort.
"We have to rest and we have a very tough game against Las Palmas (on Saturday)," said Diego Simeone, refusing to be drawn on whether his team were still hopeful of catching up with Barcelona and Real Madrid.
His son was equally determined to follow the party line.
"We have to keep working, enjoy this win and then think about Las Palmas," said Giuliano.
Defeat leaves Valladolid 14 points from safety with seven games remaining, and their fate will likely be sealed by the end of the month.
