The Premier League champions endured relentless pressure from a vibrant Gunners side during the first half at Wembley and Arsene Wenger's men were duly rewarded by goals from Santi Cazorla and Aaron Ramsey.
David Silva's half-time introduction gave City fresh impetus as Stevan Jovetic headed against the post but Olivier Giroud's dipping strike from 25 yards settled matters.
Pellegrini told BT Sport: "During the first 45 minutes, Arsenal scored two goals. We improved in the second half, we had chances to score, but they had a chance and scored from it.
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"I think, in the Premier League, you have five or six teams who can win the league because all have a strong squad. We must demonstrate from the beginning why we won the title last year."
Without captain Vincent Kompany, Pablo Zabaleta, Martin Demichelis and new signing Bacary Sagna due to the quartet's late return from World Cup commitments, City's makeshift defence endured a torrid afternoon.
It is an area of his squad that Pellegrini hopes to strengthen over the coming days with the signing of centre-back Eliaquim Mangala from Porto - a transfer saga that has dragged on throughout the close-season - and he is confident of getting the deal over the line.
"Some papers are maybe missing but we continue having interest in the player," the Chilean explained. "He wants to come so we will see what happens in the week."
In Kompany's absence, Yaya Toure took the armband for City - returning to competitive action following a close-season during which his future in Manchester was thrown into doubt by bizarre revelations from the player and his agent expressing unhappiness with how the club marked his 31st birthday.
Toure - who also suffered the tragic loss of his younger Ibrahim to cancer in June - was a goalscorer when City lifted the Community Shield with a 3-2 win over Chelsea in 2012 and he pointed to their subsequently lacklustre title defence as a reason not to read too much into Sunday's result.
"It's quite difficult because we don't have all the lads back from the World Cup but it's [good for] the fitness for the beginning of the season," he told City's official website.
"We need to work hard and bounce back against Newcastle because it's going to be a very, very important game."