Atletico had taken the lead late in the first half when Suarez teed up a deflected goal by Marcos Llorente and the task should have got easier for the visitors when Alaves defender Victor Laguardia was sent off in the second half.
Atletico would not be undone, however, and they clinched a vital victory for their title charge when substitute Saul Niguez picked out fellow late addition Joao Felix, who cut the ball across goal and Suarez pounced from less than a metre out to bundle it into the net.
Atletico went into the match trailing Real Madrid by a point but the last-gasp victory took them on to 38 points, two ahead of their city rivals and with two games in hand.
Alaves are 13th on 18.
"We knew it would be a difficult game as Alaves always make things tough and they proved that by drawing level with one player less but it was so important for us to get the win," said goalscorer Llorente.
"Our substitutions worked very well and we got the second goal after a great piece of play and it allows us to keep focusing on our objective."
Atletico had been boosted by the news Kieran Trippier's 10-week suspension from the English Football Association for violating betting rules had been put on hold by world organising body FIFA, although the right back did not travel to the match as the squad had been named before the announcement was made.
Sime Vrsaljko started in Trippier's place while record signing Felix began the game on the bench but was introduced after the hour mark and the Portuguese played a decisive role in the win, with his incisive late pass to the feet of Suarez.