Aubameyang was one of six players recalled as Arsene Wenger rotated ahead of next week's return leg with the Italians, and had a golden opportunity to score inside three minutes. The Gabonese striker was released by Mesut Ozil's slide-rule pass but saw his powerful shot blocked by Oscar Karnezis.
Arsenal's opener came five minutes later when Ozil swung in a free kick which Mustafi directed back across Karnezis and into the top corner, after being allowed to run in between two Watford defenders.
The visitors had a couple of first-half chances to equalise as Cech saved from Abdoulaye Doucoure, and Roberto Pereyra blazed over the crossbar after the goalkeeper had parried a Richarlison free kick.
Cech dived low to his left to keep out a header from the Brazilian in stoppage time as Watford grew into the game, and they continued this momentum into the second period only to be hit by a sucker-punch.
Mkhitaryan timed his through-ball perfectly and Aubameyang kept his composure to round Karnezis and slot home on 59 minutes, for his third goal in six matches since joining the Gunners in January.
Aubameyang later returned the favour to his former Borussia Dortmund teammate as he teed up a left-footed shot which went through Karnezis, who may have been blinded by Adrian Mariappa's attempted block.
Watford had previously wasted a great opportunity to mount a comeback when Cech saved Deeney's central penalty, after Ainsley Maitland-Niles had tripped Pereyra two minutes after Arsenal's second goal.
The fight that the Hornets had showed to match their hosts for an hour disappeared with the third goal as they slipped to just a second defeat under new manager Javi Gracia, although they remain in the top half of the table.
Arsenal are still 10 points adrift of the top four but appear to be getting back on track after a difficult run.
ARSENAL (4-2-3-1): Cech; Maitland-Niles, Holding, Mustafi (Chambers, 73'), Kolasinac; Elneny, Xhaka; Iwobi (Welbeck, 66'), Ozil, Mkhitaryan (Wilshere, 78'; Aubameyang
WATFORD (4-2-3-1): Karnezis; Femenia (Hughes, 62'), Prodl, Mariappa, Holebas; Doucoure, Capoue; Janmaat (Britos, 80'), Pereyra (Okaka, 67'), Richarlison; Deeney