Gunners manager Arsene Wenger had failed to land any major summer transfer targets, but that did not stop his side racing into an early lead through Olivier Giroud.
However, Villa - who spent much of last season battling against relegation - hit back when Christian Benteke equalised from a penalty rebound.
The visitors were awarded a controversial penalty on the hour when Koscielny looked to have won the ball in his challenge with Gabriel Agbonlahor, before Benteke netted from the spot.
The France defender, booked for his challenge by referee Anthony Taylor, was then soon shown a second yellow for clattering into Andreas Weimann
Arsenal, who face a crucial Champions League play-off first leg away to Fenerbahce next week, rallied with 10 men, but were not able to break down a determined Villa defence, as goalkeeper Brad Guzan made a couple of smart saves.
Antonio Luna added a late third for the visitors before unhappy chants of "spend some money" rang out from the home support at full-time.
The match was also notable for featuring the first use of the Premier League's new goal decision system (GDS) when a 52nd-minute effort from Aston Villa midfielder Fabian Delph struck the inside of the right-hand post and rolled across the line.
GDS came into play as television replays shown in the press box showed the referee Taylor had been notified that the ball did not cross the line at any point.
Arsenal had started brightly and it was no surprise when they swept into the lead on six minutes.
Source: PA
Source: PA