Bayern s Robert Lewandowski controls the ball during the German Bundesliga soccer match between FC Cologne and Bayern Munich in Cologne, Germany, Saturday, Jan. 15, 2022. AP
The reigning champions restored their six-point advantage after Borussia Dortmund had cut the gap with a 5-1 thumping of Freiburg on Friday.
Visitor's Bayern, almost back to full strength after nine players missed last weekend's home loss to Borussia Moenchengladbach following positive Covid tests, raced into a 2-0 lead in Cologne with goals by Lewandowski and Corentin Tolisso.
"After the defeat against Moenchengladbach, it was important for us to get our first points this year," Neuer told Sky.
Lewandowski reached the 300 mark by netting twice in the second half, with both assists provided by substitute Leroy Sane.
Only Bayern and West Germany legend Gerd Mueller, who died last year, scored more Bundesliga goals with 365.
"We are happy and grateful to have Robert - he is a machine up front," added Neuer.
Lewandowski opened the scoring with some slick finishing before Tolisso doubled Bayern's lead by thumping his shot into the top corner from 16 metres out.
Cologne looked to have clawed a goal back through Mark Uth, but the VAR spotted he was offside.
The scored stayed 2-0 until Leroy Sane, one of the players who missed the Gladbach loss, came on and carved holes in Cologne's defence for Lewandowski's next two goals.
Sane's pass put Lewandowski in behind the defence for his second goal.
Eight minutes later, Sane again danced through Cologne's midfield before putting Lewandowski into position to tuck away his 23rd league goal this season.
Lewandowski has scored in 15 of Bayern's 19 league games.
He is on course to break his own record of 41 Bundesliga goals in a single season, which he set last May.
Union Berlin went fourth, and into the Champions League places, with a 2-1 home win over third-placed Hoffenheim as Andreas Voglsammer and Grischa Proemel scored for the hosts after an own goal by team-mate Timo Baumgartl.
RB Leipzig continued their climb, rising to seventh with a 2-0 win at VfB Stuttgart as Portuguese striker Andre Silva bagged his fifth goal in as many league games.
Silva, who also scored a brace in last weekend's home win over Mainz, converted an early penalty after a handball by Stuttgart defender Konstantinos Mavropanos.
Leipzig doubled their tally with a superb counter-attack from their own box when Benjamin Henrichs put Christopher Nkunku away and the French winger fired home with 20 minutes left.
Wolfsburg ended their run of six straight league defeats with a goalless draw at home to Hertha Berlin, but both clubs remain just above the bottom three.
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