The Portuguese team started matchday six two points in front of third-placed Schalke, but were knocked out after their defeat was coupled with the German side's 1-0 success at Maribor.
Marco Silva's men made a terrible start to the clash as a Cesc Fabregas penalty was followed by Andre Schurrle's fine effort from distance in the first 16 minutes.
Sporting cut Chelsea's two-goal lead in half in the 50th minute through an excellent Jonathan Silva strike, but John Obi Mikel's first Champions League goal gave the hosts breathing space.
And news of Schalke's goal filtered through in the latter stages to cap a miserable night for Sporting, although they will now compete in the UEFA Europa League.
Chelsea's win ensured they finished the group undefeated, with four wins and two draws - giving them six more points than Schalke.
Jose Mourinho made six changes after Chelsea's first loss of the season at Newcastle United on Saturday as the likes of Schurrle, goalkeeper Petr Cech and highly rated defender Kurt Zouma came in.
Chelsea's next meaningful attack resulted in another goal, as Nemanja Matic - who scored in his side's 1-0 win at Sporting in September - drove forward with purpose, before finding Schurrle.
Patricio did better to keep out Schurrle's effort on the bounce in the 23rd minute, while Matic curled a fizzing left-footed strike just over after Mohamed Salah drew a swarm of Sporting defenders towards him on the left.
Sporting were neat on the ball and rarely made errors, but simply could not compete with Chelsea's speed and power.
And although they created a decent chance when the ball spilled to Diego Capel after a goalmouth scramble, Cech was alert to come off his line and block the Spaniard's strike, keeping Chelsea two goals ahead at the break.
Schurrle tried to head clear, but the ball hit his back and bounced invitingly to Jonathan Silva, who chested it down before blasting a finish into the left corner.
The goal buoyed by the travelling fans, but Chelsea's riposte was swift and devastating.
First, Schurrle played in Salah and, although his excellent effort forced a great save from Rui Patricio in the 52nd minute, Chelsea did not have to wait long for a third goal.
Fabregas swung in a free-kick from the left and Mikel nipped in front of Matic to meet Gary Cahill's flick-on and stab home from less than a yard.
Cech made a good save to keep out Islam Slimani's header, following good work from Carlos Mane down the right, but that was as close as Sporting got to a second.
