Bayer Leverkusen moved into third place after beating lacklustre Schalke 04 1-0 thanks to a Sidney Sam goal.
Champions League competitors Schalke, who have failed to win any of their last five games, dropped back down to 17th spot with six points from 10 games.
Eintracht Frankfurt continued their fine run with a 3-1 win at St Pauli - their fifth victory in six games - to climb to fourth place on 16 points. Frankfurt's Fanis Gekas struck twice to go top of the scorers' list with nine goals from 10 games.
Slovenian Novakovic netted twice in 18 first-half minutes to put Cologne ahead and then bring them level after Hamburg had taken a brief 2-1 lead through Son Heung Min.
Paolo Guerrero rattled the crossbar early in the second half for a superior Hamburg before Novakovic completed a swift move six minutes from time to seal a victory that lifted Cologne off bottom spot and into 15th place.
Schaefer, who had replaced the sacked Zvonimir Soldo as coach last week, celebrated his first win in his first Bundesliga game after also steering Cologne to a 3-0 win over 1860 Munich in the German Cup in midweek.
SCHAEFER STAYS
Immediately after the game Cologne said they would stick with caretaker Schaefer for now.
"I am delighted we have won this crucial game for the club because if you look at the table you realise how big a game that was for us," Schaefer said.
Hamburg dropped to eighth place on 15 points.
Mercurial Werder Bremen dropped to 11th place after losing 3-2 to visitors Nuremberg.
VfL Wolfsburg celebrated their first win in four games when they beat VfB Stuttgart 2-0 with goals in either half from captain Edin Dzeko to move up to 12th spot on 13 points from 10 games.
Dzeko's first goal brought a new Bundesliga scoring record for home games, with coach Steve McClaren's team having now scored in their last 40 home matches, breaking Bayern Munich's record of netting in 39 between 1986 and 1989.
Bayern eased past Freiburg 4-2 on Friday to move up to seventh place on 15 points.
