Baumgartner opened the scoring just after ten minutes before Cologne were sent down to ten man in the 26th minute, after Sebastiaan Bornauw received a straight red card.
Baumgartner then headed home a second a minute after the restart of the second half, and then in the 48th minute, he set up left back Steven Zuber to score a third.
Two minutes later Hoffenheim themselves went down to ten men after Benjamin Hubner picked up a second yellow card, and Cologne did try to find a way back into the game when Florian Kainz smacked the ball past Michael Baumer on the hour mark.
Although, Markus Gisdol’s side were not able to form a comeback, even though they did have a chance to make it an interesting last 15 minutes from the spot. However Mark Uth's effort was kept out by Baumann.
Hoffenheim's ideal first half
It was the visitors that made the brighter start to the encounter, but Hoffenheim broke the deadlock inside 12 minutes when Baumgartner slotted home from Jacob Brunn Larsen’s first time low cross from the left.
Then on the 26th minute mark, Cologne were sent down to ten men when centre back Bornauw was sent off for a late challenge on Baumgartner when coming in behind the midfielder with his studs up.
Originally the referee deemed the challenge to be a yellow card offence, but after going to the VAR screen, he reversed his decision to a straight red card.
Despite having a man less, the visitors went close to getting an equaliser with eight minutes of the first period remaining, when Jhon Cordoba teed up Elvis Rexhbecaj inside the penalty area, but his low drive was deflected narrowly wide of the far post.
Two minutes later Ismail Jakobs had an effort cleared off the line when Ellyes Skhiri cut the ball back to the winger, but his low shot was fantastically hooked away to safety by the hosts.
On the 43rd minute, Alfred Schreuder’s team thought they had doubled their advantage when Munes Dabbur was played through on goal, who squared the ball to Baumgartner to tap home. However, the goal was chalked off by VAR for offside against Dabbur.
Then in first half stoppage time, Dabbur maybe felt that he should have scored himself when his header from point blank range was kept out well by the legs of Timo Horn.
Baumgartner on form
After Hoffenheim's strong end to the first half, less than a minute after the restart, Baumgartner doubled his tally and his sides lead after he headed past Horn from Robert Skov’s cross from the right, despite efforts from Jorge Mere to clear the ball off the line.
The hosts then added a third when Baumgartner neatly flicked the ball into the path of Zuber who slid the ball into the bottom right hand corner.
However, two minutes later the home side went down to ten men themselves when centre back Hubner received a second yellow card for a late challenge outside of the box on Dominick Drexler.
Cologne attempt at a comeback
The away side did get a goal back on the hour mark when Cordoba flicked the ball to substitute Florian Kainz, who via a deflection smashed the ball past Baumann.
On the 76th minute mark Cologne were then awarded a penalty via VAR after the referee deemed the ball to hit substitute Sebastian Rudy’s hand inside the area.
Although Gisdol's team failed to convert the spot kick as Mark Uth’s effort was kept out by Baumann after he guessed the right way.
What it means for both sides