The only negatives in a comfortable win for United were injuries to Mkhitaryan and midfielder Michael Carrick, who could both miss Sunday's English League Cup final.
United will be joined in the next round by Schalke 04, whose 1-1 draw in their last 32 second-leg tie with PAOK Salonika was enough to secure a 4-1 aggregate win, and FK Krasnodar who went through against Fenerbahce.
Krasnodar drew 1-1 on the night to advance 2-1 on aggregate, with the remaining last-32 ties to be played on Thursday.
United manager Jose Mourinho played a strong side including Zlatan Ibrahimovic who scored a hat-trick in last week's 3-0 win over the French side and it did not take them long to effectively kill off the tie.
Juan Mata was given acres of space on the left side of the pitch and the Spaniard clipped a superb low cross into the box for Mkhitaryan to get in front of his marker and flick a low finish into the net.
That left St Etienne needing five goals, a mountain they showed little inclination to climb even after Bailly was sent off in the 63rd minute after picking up two yellow cards in the space of three minutes for fouls on Romain Hamouma.
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Manchester United's Ashley Young vies for the ball with St Etienne's Henri Saivet Reuters
Mkhitaryan departed with a hamstring problem and Carrick suffered a calf injury, meaning both are doubtful for Sunday's Wembley clash against Southampton.
Mourinho, however, believed it was the right decision not to rest more first-team players ahead of the cup final.
"You can be injured leaving your house or in the garden, you never know why, you always have the risk of it," Mourinho told BT Sport.
"If someone is missing, it is opportunity for another, although not having both for the final is complicated."
Schalke were in pole position against PAOK with a 3-0 first-leg lead and tightened their grip on the tie after 23 minutes when Alessandro Schoepf scored for the second game in succession.
PAOK drew level two minutes later when Matija Nastasic diverted the ball into his own net, but there was no further drama as Schalke comfortably saw the game out.
Krasnodar had won 1-0 at home against Fenerbahce and doubled the aggregate advantage when Fedor Smolov scored with the visitors' first shot of night after seven minutes, before Fenerbahce drew level when Brazilian Souza poked in after a goalmouth scramble.
St Etienne 0 Manchester United 1
Souza 41 FC Krasnodar 1
Fedor Smolov 7
Schalke 04 1
Schoepf 23 PAOK Salonika 1
Matija Nastasic 25(og)