The two teams, who both had players sent off in the pre-season affair, could not be separated at the end of 90 minutes, with a 2-2 draw ultimately bringing a penalty shootout.
Bale could not add to his second-half goal by converting from the spot, but Zinedine Zidane's side still ran out 3-2 winners in the shootout to pick up their first victory of the summer.
It did not take long for Arsenal to find their feet, though, with the Gunners making the breakthrough in the 10th minute courtesy of a penalty from Alexandre Lacazette.
Nacho was fortunate to stay on the field minutes before the penalty incident for a rash challenge on Sead Kolasinac, but the Madrid centre-back did receive his marching orders for blocking Lacazette's goal-bound attempt with his arm.
The France forward saw his spot kick hit both posts before dropping into the back of the net and Unai Emery's side quickly went in search of their second with Henrikh Mkhitaryan forcing a smart save from Keylor Navas.
Arsenal doubled their advantage in the 24th minute, though, with Pierre-Emerick Aubameyang racing onto a pass from Lacazette before rounding Navas and finding the back of the net.
Luka Modric struck over Martinez's crossbar in the 28th minute as Madrid looked to answer back, but it was a struggle for the Spanish giants having been reduced to 10 men at such an early stage.
Lacazette had another chance to make it three for the Gunners in the latter stages of the half, but the France international struck over the crossbar from a dangerous position.
There was to be more drama late in the first period, though, with Arsenal also being reduced to 10 men as Greek defender Sokratis Papastathopoulos picked up two yellow cards in quick succession - the second of which arrived for a late challenge on Modric just outside the penalty box.
Madrid should have halved the deficit in the final moments of the first half when Toni Kroos picked out Benzema inside the Arsenal box, but the striker somehow nodded his effort wide of the post.
Bale was amongst the Madrid changes at the interval, with the Welshman being sent onto the field despite the widespread speculation surrounding his future.
Marco Asensio - another of the changes - came close to registering in the 52nd minute when he struck a brilliant effort towards the corner of the goal, only for the post to come to Arsenal's rescue.
Los Blancos were on the scoresheet just before the hour, though, with Bale converting from close range following a scramble inside the Arsenal box.
Less than three minutes after Bale's strike, Madrid levelled the scores through Asensio, who combined with Marcelo in a dangerous position before beating Martinez at his near post.
Asensio had to be stretchered off the field not long after, though, with the Spain international appearing to suffer a knee problem.
Martinez had to keep out a clever attempt from Bale in the period that followed as Madrid continued to force the issue against an Arsenal team that looked tired in the second half.
Bale did excellently to clear a Calum Chambers effort off the line in the 76th minute, moments after Emery had looked to his bench and made four alterations. The Madrid attacker then curled one just wide of the post down the other end as both teams went in search of a winner.
Eddie Nketiah came close to winning it for the Gunners late on after breaking into a one-on-one situation with Thibaut Courtois, but the youngster saw his attempt saved as the match headed for penalties.
Bale missed Madrid's first effort in the shootout, but Isco, Varane and Vinicius Junior all registered for Zidane's side, who won 3-2 on penalties with Granit Xhaka, Nacho Monreal and Robbie Burton missing for the Gunners.
Next up for Madrid is a clash against bitter rivals Atletico Madrid in New Jersey on Saturday, while Arsenal will take on Lyon in the Emirates Cup on Sunday.
REAL MADRID FIRST-HALF XI: Navas; Carvajal, Ramos, Nacho, Mendy; Vazquez, Modric, Kroos, Hazard; Jovic (Varane 16'), Benzema
REAL MADRID SECOND-HALF XI: Courtois; Odriozola, Varane, Ramos (De La Fuente 65'), Marcelo; Isco, Kroos (Fidalgo 65'), Modric (Seoane 65'), Asensio (Vinicius 65'); Bale
ARSENAL: Martinez, Chambers, Sokratis, Kolasinac; Mkhitaryan (Nelson 75'), Willock (Monreal 46'), Xhaka, Ozil (Sako 75'), Jenkinson; Aubameyang (Burton 75'), Lacazette (Nketiah 75')