It was a club-record 15th straight league home win for Atletico, surpassing the run of 14 victories in a row at the Vicente Calderon Stadium between May 2012 and February 2013.
Atletico fell behind when striker Gerard Moreno scored in the 20th, but the hosts rallied with goals by Axel Witsel in first-half added time, Antoine Griezmann in the 80th and Samuel Lino in the 85th.
It was the seventh goal in six matches for Griezmann who, along with Alvaro Morata, is leading an attack that has scored 29 goals in 12 league matches.
Atletico were coming off a 6-0 home rout of Celtic in the Champions League and are unbeaten in 12 games in all competitions, with 10 wins and two draws. Atletico have won all of their six games at home this season, plus the last nine from last season to set the new club record at the Metropolitano stadium.
"It's always a plus to play in front of these fans," France international Griezmann said.
"They always give us an extra boost."
The team coached by Diego Simeone, who last week had his contract extended until 2027, have a game in hand as a home match against Sevilla in September was suspended because of bad weather.
Fellow Catalan club Girona had also come from one down to win 2-1 at Rayo Vallecano, also on Saturday.
Alaves took the lead in a breakaway one minute into the match, but Lewandowski helped the hosts rally with a header in the 53th and a penalty in the 78th.
Lewandowski, who was sidelined recently with an injury, hadn't scored in six matches since getting a double in the 3-2 win against Celta Vigo in September.
"We played a bit better, but we are not going through our best moment," Barcelona coach Xavi said.
"The international break will be good for us so we can regroup. We will improve, I'm certain."
Barcelona extended their unbeaten run against 14th-placed Alaves in the league to 12 league games, with 10 wins and two draws.
Ayoze Perez had put the visitors ahead in the 72nd.
Seville, sitting in 13th place, had their winless streak in the league extended to six matches, while seventh-placed Betis had won five in a row in all competitions.