Besiktas spurned clear chances to take the lead, with Mehmet Akgun and Manuel Fernandes both missing first-half openings.
Aguero made the Turkish side pay on the stroke of half time, finishing a swift counter-attack with a cool left-footed finish.
Kompany sealed City's first pre-season win when he raced down-field to collect Adam Johnson’s through ball and majestically lifted the ball over goalkeeper Cenk Gonen.
In the early evening rain in Innsbruck, conditions were not helped by the decision to debut City’s new burgundy away kit against a Besiktas side playing in red.
And the colour clash seemed to affect the Manchester side more than the Turks, who created the best chance of the game early on.
Mehmet couldn’t keep his shot down, however, as he looked to guide a shot under the bar from the corner of the box with Costel Pantilimon out of position.
But the game hinged on a big miss by Fernandes, who took up a superb position in front of goal – but side-footed Mehmet’s cross straight at Pantilimon, who got up well to smother Olcay Sahan's rebound.
That glaring miss seemed to kick-start City into action, with Carlos Tevez looking particularly lively in attack. And the Argentine showed signs that his head could finally be in the right place ahead of the new season, as he played a key role in City’s opener.
Yaya Toure led City’s counter-attack and found Tevez wide on the left. The striker drifted inside and spotted the run of Aguero, who collected the ball after Abdul Razak’s intelligent dummy, and steered a classy finish into the bottom corner.
That strike, moments before the break, knocked the stuffing out of Besiktas, who had acquitted themselves well against the Premier League champions until that point.
But any hopes of salvaging something from the game were dashed when Kompany’s rare goal put the game beyond the Turkish side.
Striding forward from defence, Kompany’s run was spotted by Johnson’s pinpoint pass – and the Belgian skilfully chipped the ball over the advancing Gonen, whose hand to the ball couldn’t prevent it from dropping under the bar.
Mustafa Pektemek could have reignited the contest soon after, but his low drive against the base of the Pantilimon’s left-hand post was as close as Besiktas came.
A flurry of second half changes took the heat out of the contest, but it was a competitive workout for both sides – with City’s next outing against Arsenal in Beijing’s Bird’s Nest stadium next Friday.