Danny Welbeck scored the only goal of a forgettable game as he turned home a 77th-minute winner for the visitors.
Both sides started the game slowly and the first half was largely bereft of quality, with Emery’s side gradually improving after the interval as Welbeck secured the 1-0 win.
Since losing to Premier League champions Manchester City and London rivals Chelsea in his first two games in charge, Emery has tasted nothing but victory.
It is 11 years since Arsenal went on such a winning run, with the club’s all-time record streak standing at 14 matches.
Premier League victories over Crystal Palace and Liverpool, as well as Carabao Cup success against Blackpool in-between, would see Emery’s team draw level with their counterparts from 1987, but the Spaniard is not looking too far ahead.
“Three matches, that is a lot of matches,” he said.
“We are only thinking of the next match against Crystal Palace.
“Today we were with them and they were with us, supporters who had come from Arsenal, from London or from here. For us, it’s very important that supporters are with us and enjoy it.
“I’m enjoying continuing the work, our process. They have a high demand for us because every match is not easy for us.
“We are thinking about three points at Crystal Palace but we know that it’s very difficult to win there.
“For us, continuing to win is what we want but we know it’s not easy and we can lose.”
Despite the current run of form, one criticism of Arsenal has been their slow start to matches, which was evident again at the Estadio Jose Alvalade.
Emery concedes it is an issue that needs addressing and that he has made his players aware of it.
“We spoke in the dressing room before the start of the match today about our objectives. And one objective is this.
“But we need to continue finding the solution for us to be better in the first halves.”
Sporting boss Jose Peseiro felt his team were on the receiving end of some poor officiating, but spoke with pride about his players.
“Nobody has been able to do what we did in that first half against Arsenal,” he said.
“I’m not sure whether it was lack of aggression or fear but they came back into the game. We were playing one of the best sides in Europe, who play very well vertically and with great depth. I cannot blame my team for doing what they did.”