The Colombian poked home a shot from Raul Garcia with 14 minutes remaining to decide a closely-fought match that was high on intensity and tension but which produced little in the way of quality football.
The last match between these two sides ended with two Sevilla players being sent off and, although no players were dismissed this time around, the match was no less heated and Atletico's players picked up five yellow cards between them - the most notable being handed to Mario Suarez for his thunderous challenge on Jose Antonio Reyes.
Neither side managed a single shot on target in a tight first half.
Suarez and Costa had Atletico's best efforts from distance, while Negredo shot over the crossbar from Sevilla's most promising opening.
The opening minutes of the second period saw more goalmouth action than the entirety of the first.
First Diego Godin cleared a dangerous cross from Jesus Navas, then Coke forced a brilliant save from Thibaut Courtois by heading a cross from Navas towards the bottom corner, but the Belgian goalkeeper got down low to make the save.
Diego Costa was awarded a yellow card for a challenge on Geoffrey Kondogbia which means he will miss next week's Madrid derby, and the Brazilian further enraged the home crowd by feigning injury following a confrontation with Juan Cala, trying to make it look as if the centre-back had head-butted him.
Coach Diego Simeone sensibly withdrew the forward and replaced him with Adrian Lopez.
Atletico finally broke the deadlock on 76 minutes.
Suarez fought to win the ball in the air on the edge of the area and switched it to Adrian on the left hand side of the box.
The midfielder then delivered a brilliant ball to the far post, which Garcia volleyed towards the far corner and Falcao helped it into the net with his studs.
Sevilla's players, however, were adamant that Suarez had used his arm to control the ball in the build up and were furious that the goal was allowed to stand.
Unai Emery's side finally sprang into action in the final moments of the game.
Courtois was forced to save two long-range efforts from Kondogbia and then a far-post header from Alvaro Negredo. The striker then missed his side's best chance of the game from the resulting corner, heading just wide of the far post from only a few yards out.
There was still time for one more nervy moment for Atletico, with Ivan Rakitic smashing the top of the bar from a free-kick in stoppage time.
The win means Atletico have narrowed the gap between themselves and second-placed Real Madrid in the Primera Division table to just three points ahead of next weekend's derby, while Sevilla slip down to 11th.