After heading the winning goal, Dragovic ran along the touchline making an aggressive gesture with his arm before celebrating with the Basel bench.
The Austrian, given a second yellow card after the referee interpreted his behaviour as a provocation to the home supporters, was sent off amid chaotic scenes, with his team mates having to restrain him.
"I had a bet with our assistant coach. If I scored with my head, I would have to run around the field," he told reporters.
"The gesture was aimed at him, not the Servette fans."
Servette had Omar Kossoko dismissed for throwing the ball down in disgust at a decision and getting a second yellow card.
Last year, Dragovic had to apologise for playing a schoolboy prank on a Swiss minister during the medal ceremony at the end of the domestic Cup final.
Dragovic gently slapped the balding politician on the back of the head as he was handing out medals to other Basel players.
Sunday's win took second-placed Basel, champions for the last three seasons, within two points of Swiss Super League leaders Grasshoppers who drew 0-0 at Lausanne-Sport.
On Saturday, former AC Milan and Italy midfielder Gennaro Gattuso's first league match as FC Sion player-coach ended in a 3-1 defeat at FC Zurich.
Gattuso, who joined Sion as a player in July, is the club's fifth coach of the season.