Signs of rustiness were evident on Tuesday night as the Abu Dhabi side beat Sharjah's city rivals Al Shaab thanks to a late penalty.
Spanish manager Milla was happy with the result, but not the performance.
"The first match is always hard but we managed to get the result we wanted, which was important for the moral of the team," he said. "Now we are not only looking to take the winning mentality forward but combine it with improvement, particularly being the first home game and in front of our home fans.
"We had a few areas to iron out from the previous game and in this match we will show how much we have improved. I can't promise more goals but a better performance, for sure."
Jazira will be without their €10 million (Dh47m) summer signing Abdelaziz Barrada for Sunday night's home encounter as the Moroccan midfielder was on national duty for the World Cup qualifier against Ivory Coast on Saturday night.
Striker Ali Mabkhout, midfielder Khamis Ismail and goalkeeper Ali Kasheif are all in Saudi Arabia with the UAE team for the OSN Cup, a four-team friendly tournament where the Emirates face New Zealand in Monday's final.
Milla has said he will also adopt a rotation policy for cup games this season to try keep his players fresh and give experience to fringe players.
"We are looking at a long season ahead," he said. "We will play in three domestic competitions and the Asian Champions League."
The match sees Paulo Bonamigo, the Sharjah coach, face off against his former club for the first time since his dismissal.
The former Al Shabab manager started last season with Jazira but lost his job in February and was replaced by Milla.
Milla expects Bonamigo's side to be strong opponents.
"Sharjah will be highly motivated team for many reasons," Milla said. "They have a coach who knows the Jazira players quite well and they also want to underline their status in the top flight after the promotion.
"At the same time, I have full confidence in my players to show the right spirit and attitude and fight hard to take the three points of this match."
Musallem Fayez, the defender who arrived at Jazira from Al Ain as part of the deal that saw Ibrahim Diaky leave the Mohammed bin Zayed Stadium, is also expecting a vastly improved performance from the team.
"Our performance in the first match was not satisfying due to many factors, but I promise that that result is behind us and for our fans to expect 200 per cent from all the players," he said.
AJMAN v AL SHABAB
Abdul Wahab Abdul Qader's side came from a goal down to clinch a 2-1 result over Al Nasr.
Shabab will be looking to bounce back after a 1-0 home defeat to Al Wasl.
Shabab had to finish their first game with 10 players after Dawood Ali was sent off in the 27th minute.
"My team still played well, the new players are doing well and I am confident of a positive result," manager Marcos Paqueta said.
DUBAI v AL SHAAB
Correa, the former Al Nasr forward, will combine with compatriot Andre Alves in attack.
Marius Sumudica, Shaab's Romanian coach, is not taking the League Cup lightly.
"We have strengthened the squad and obviously we are looking for good results from the beginning of the season. For us it is another big game," he said.
PICKS CAN ADD UP TO PRIZES IN FANTASY LEAGUE
The Arabian Gulf League has launched its 2013/14 fantasy football game with prizes on offer for top-scoring managers including iPad minis.
Fans can register free of charge at www.fantasy.AGLeague.ae and select their squad ahead the first games of the weekend on September 14.
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