Elneny turned in one of his best performances yet for Arsenal to help Arsene Wenger's side remain in the contest after they fell behind to a Victor Moses goal at the start of the second half.
The Egyptian explained that motivation was high within the squad to make the most of their early entry into competitive action.
"We had the motivation to win the game against the league winners - Chelsea - and especially [because it was] almost a week before the start of the league," Elneny said after the match.
"Winning the game will give us such a great push forward before the start of the new season.
"It will help us to start it in the best possible way."
Wenger agreed with his midfielder's assessment but stressed his side must maintain their standards.
"We want to find a good balance between the confidence we can get out of these games and the urgency that every Premier League game demands," the Arsenal boss said.
"Let's go into the Premier League with the same discipline, the same spirit, and see where we can go."