More than 60,000 fans cheered at a packed Estadio da Luz when Spanish midfielder Garcia headed what proved to be the winner just before half-time, beating Stijn Schaars in the air as he moved in on a Pablo Aimar corner.
Benfica's Paraguay striker Oscar Cardozo was sent off just past the hour-mark but the hosts dug in to defend their slender lead until the end.
"It was a very intense match, extremely contentious. We ended up earning an edge on a set piece and then knew how to suffer in the last half hour when we were left with 10-men," said Benfica coach Jorge Jesus.
Benfica now have 27 points from 11 games. Sporting, who remain third with 23 points, started well with midfielders Schaars and Elias pressing high up the pitch and creating chances but they were progressively dominated by Benfica's combative duo of Garcia and Axel Witsel.
Silky playmaker Aimar put on an impressive performance and helped Benfica win the midfield, switching defence to attack and linking especially well with winger and felow Argentine Nicolas Gaitan.
Before the goal, Gaitan hit Rui Patricio's left-hand post with a stunning first-time shot from outside the box, after Aimar's floating corner, before Schaars almost put Sporting ahead with a volley that flew inches wide.
"It was a great match, a spectacular one, the draw would have been fairer but we lacked enough focus in the end," said Sporting coach Domingos Paciencia.
In a match with plenty of chances for both sides, the game was evenly balanced until Cardozo got a second yellow card for dissent after 63 minutes.
"For the last 25 minutes we suffered a lot but pulled through. The fans were amazing and we needed that more than ever," said match-winner Garcia.
Sporting took control after the sending off and should have equalised but Elias failed to put away an easy chance and the hosts held out for a precious win to cap a perfect week after their 2-2 draw against Manchester United at Old Trafford on Tuesday put them into the last 16 of the Champions League.
