Santi Cazorla had an impressive debut for the London side but another recent arrival, striker Olivier Giroud, missed a great chance to score his first goal for the club and seal the victory with 10 minutes remaining.
The stalemate was the third time in as many years that the Gunners have begun their campaign by only picking up one point, and it is also the third goalless draw in the last five Premier League games at the Emirates.
With their departed captain and top scorer preparing to make his Manchester United debut on Monday, Arsenal fans were given their first competitive glimpse of the future of their attack. New signings Lukas Podolski and Cazorla both started with their third summer acquisition, Giroud, on the bench.
The presence of those new star names on the teamsheet was cause for optimism among Arsenal fans, but that was more than offset by the absence of midfielder Alex Song from the matchday squad amid reports that a move to Barcelona is in the offing.
Arsenal completely controlled the first half, enjoying 70 per cent of the possession, but it was Sunderland who had the first chance of the game after five minutes. James McClean was played into the box on the left and his shot from an angle was blocked by Wojciech Szczesny. Arsenal immediately countered, and Cazorla galloped forward before firing in a shot from 20 yards that Simon Mignolet parried with an outstretched arm.
That was the first significant action of a highly influential debut for Cazorla. The Spaniard struck a low shot just wide after both he and Podolski went to meet Gervinho's cross, crossed for Theo Walcott to head over the bar and generally provided a good link between midfield and the front three.
Abou Diaby, starting in the missing Song's stead, forced a good low save from Mignolet with an effort from range, but they ended the half with just two shots on target, the same as Sunderland.
The second half followed much the same pattern as the first, with most of the action in the Sunderland half as the visitors did their best to keep Arsenal at bay.
Cazorla volleyed wide after one of many searing runs into the box from Gervinho saw a deflected shot fall for him, before Theo Walcott hit the side-netting at the end of another promising move.
Midway through the second half Arsenal boss Arsene Wenger replaced Podolski – who had been tightly marked throughout – with Giroud. The French striker, who finished joint top-scorer in Ligue 1 last season as Montpellier won the title, did well to connect with Karl Jenkinson's cross with a side-footed volley on the turn, but he could not guide it on target.
If he could have been forgiven for that miss, then his next chance will have made the realisation of life without Van Persie all the more sobering. With 81 minutes on the clock, Giroud timed his run well to meet Cazorla's through ball, but with only Mignolet to beat he fired wide of the target.
That miss was to prove costly, as the final whistle came without any more opportunities as clear as that, and Arsenal finished the day having not won any of their last four home games.