Two first-half strikes from Youssouf Fofana and Eddie Nketiah meant that the two sides could not be separated after 90 minutes in the British sunshine, but Takumi Minamino was denied from the spot by Aaron Ramsdale as the Gunners won the friendly tournament for the seventh time.
An experimental Arsenal XI saw Declan Rice and Jurrien Timber handed their unofficial home debuts, while Emile Smith Rowe and Fabio Vieira were also given opportunities to impress from the off.
However, the hosts' unfamiliar lineup looked disjointed and lacked rhythm during the opening exchanges, and Monaco needed just eight minutes to test Ramsdale, who got down low to beat away an Aleksandr Golovin effort.
Arsenal showed fleeting signs of life in the final third - mainly through the lively Gabriel Martinelli - but the Brazilian was culpable for Monaco's opener in the 31st minute with a piece of passive defending.
Martinelli allowed Mohamed Camara to skip past him far too easily on the right-hand side, and the Monaco midfielder fizzed a cross in for engine room partner Fofana, who got behind the Arsenal backline and found the top corner with a stunning header.
The Gunners' defensive deficiencies were there for all to see, but Rice very nearly marked his Emirates inauguration with a dream goal in the 42nd minute, racing onto Takehiro Tomiyasu's pass before being denied a certain goal by Guillermo Maripan's excellent sliding block.
From the resulting corner, Philipp Kohn produced a fine reaction save from Tomiyasu's header, but the second time was the charm for the hosts, who levelled in the 43rd minute through Nketiah's close-range volley at the back post.
Arsenal's equaliser was merited, but Arteta's side were caught cold five minutes into the second half and were grateful to see Wissam Ben Yedder's strike hit the post before bouncing off Ramsdale and behind for a goal kick.
The changes arrived en masse with just over an hour played, and two of Arteta's substitutes very nearly combined for the game's third goal in the 72nd minute, as Leandro Trossard shot straight at Kohn before picking up the loose ball and teeing up Martin Odegaard, who fired just over the top from 20 yards.
Neither side managed to carve out any more clear-cut openings in the time that remained, leading to a winner-takes-all penalty shootout, although a 12-yard showdown would have taken place regardless of the result.
Following six perfect penalties from both sides, ex-Liverpool man Minamino went for power down the middle, but Ramsdale stayed put and stuck a strong hand into the air to thwart the Japanese attacker, giving Gabriel Magalhaes the chance to slot home the cup-clinching penalty.
Arsenal now have four days to recover before their Community Shield showdown with Manchester City on Sunday, which precedes their Premier League opener at home to Nottingham Forest on August 12.