Baumgartner put Hoffenheim in charge in a first half which saw Sebastian Bornauw sent off, and Steven Zuber was on target for the hosts as they rapidly pulled clear after the restart.
Florian Kainz got Koln on the board after Benjamin Hubner joined Bornauw in being dismissed, but Mark Uth fluffed his lines from the penalty spot to spurn any possibility of a fightback.
The win ends a seven-game drought for Alfred Schreuder's men and puts five points between themselves and their opponents.
Story of the match
The opener came with the game's first clear-cut opportunity on 11 minutes. Hubner clipped the ball forward to Jacob Bruun Larsen, who had time to slide the ball across for an arriving Baumgartner. The Austrian duly tucked it through the legs of Timo Horn for his fifth goal of the season.
15 minutes later, Bornauw needlessly complicated Koln's task. He was initially shown a yellow for a lunging tackle on Baumgartl that made an imprint on the goalscorer's calf, but after a VAR review, Felix Brych brandished a red.
But the visitors did not allow themselves to be forced onto the back foot. In fact, they came very close to equalising with two quickfire opportunities.
First, Elvis Rexhbecaj shot just wide after Jhon Cordoba's lay-off, and then Uth produced a cut-back from the byline that was begging to be converted. With Cordoba having narrowly missed it, Ismail Jakobs fired it goalwards at the back post, only for Pavel Kaderabek to clear off the line.
In these circumstances, Hoffenheim would have felt great relief when they thought they had established a cushion. Again, a direct pass from the left side of the defence looked to have been Koln's undoing, Zuber this time setting Munas Dabour into an alarming amount of space. Dabour teed-up Baumgartl for an easy finish, but was ruled offside by VAR.
In the last action of the first 45, the Israeli saw a header saved by the legs of Horn.
The game was then decided in a devastating three-minute spell at the start of the second, with Koln left dazed as Hoffenheim raced out of the blocks.
Within 30 seconds of the restart, Robert Skov picked the ball up on the right edge of the area and drove forward, lifting a cross to the back post for Baumgartner. Stretching to meet it, he floated the header over the line before substitute Jorge Mere could clear.
And moments later, the 20-year-old turned provider in a superb exchange. Larsen slid the ball into him and, with his back to goal, he produced a fantastic piece of skill to tee-up for Zuber for his first of the campaign.
Koln may have feared a rout at this point, but Hoffenheim forfeited their man advantage when Hubner earned a second yellow for a late challenge on Dominick Drexler which would ultimately force the attacker to limp off.
Just after the hour, the emboldened visitors narrowed the deficit with a direct attack of their own. Horn's launch downfield was headed by Cordoba into the path of substitute Kainz, who beat Oliver Baumann courtesy of a deflection off replacement centre-back Havard Nordtveit.
Hoffenheim very nearly responded instantly when Ilhas Bebou, on for Dabour, was denied by Horn and unsuccessfully tried to turn in Skov's follow-up.
Koln had come from two goals down to gain a point against Fortuna Dusseldorf at the weekend, and they looked capable of going one better when Nordtveit conceded a penalty with 15 minutes remaining for a clear handball when challenging Uth.
It was Uth, the former Hoffenheim forward, who stepped up to take, but he could not make his mark, thwarted by Baumann's left-handed save.
The Billy Goats had one more opportunity to make the final minutes nervy for Hoffenheim, but Rexhbecaj forgivingly slotted wide.
Baumgartner sure to attract interest
After excelling at youth level for Hoffenheim, Baumgartner is beginning to catch the eye in the top flight.
This was an outstanding display to continue his impressive 2020 form. His first goal was a simple but assured finish, but his second was far more difficult. He might even have had a hat-trick if Dabour had timed his run slightly better late in the first half.
The real highlight, though, was his assist: a drag-back and a flick to put it on a plate for Zuber. You really could watch that goal on repeat.
He has now been directly involved in 10 goals in 20 Bundesliga appearances in 19/20. If he maintains this level, he will not find himself short of suitors.