The Portuguese blamed fatigue for the injuries sustained by Paul Pogba and Daley Blind in the Europa League Round of 16 second leg win over Rostov.
The two sides drew 1-1 in the first leg, so Manchester United has the edge coming home due to its away goal.
The world's most expensive player bore the brunt of the criticism as United exited the FA Cup at the hands of Chelsea.
But Mata finished off a fine United move to punish a Rostov error and that goal proved to be enough, though Sergio Romero was forced into a superb late stop to prevent the match going into extra time.
Rostov know that due to the current aggregate score they will need to score in Manchester and there is a clear sense that their best chance of eliminating United would have been with a triumph in Russian Federation, where Ajax and Bayern Munich had already been beaten this season.
"A lot of people might say we should have scored more goals".
With the Reds bossing both possession and territory, it seemed only a matter of time before the breakthrough would come.
The Armenian nearly got his revenge shortly after, but he clipped a delicate effort agonisingly wide of the left-hand upright after another fine Mata pass. We won't be playing one match a week like some of them have since the start of the season.More news: Juve through after penalty, red card end Porto hopes
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United suffered a blow at the start of the second half as Pogba was withdrawn with an ankle injury and replaced by Marouane Fellaini.
At the other end, Antonio Valencia's cross picked out Ibrahimovic, who got in front of Cesar Navas to guide a header over the crossbar.
That was until the 70th minute, when United finally took the lead.
United will find out on Friday who they will play in the quarter-finals, with the draw taking place at 12:00 GMT.
Manchester United will now turn their attentions back towards Premier League affairs and their home clash against manager-less Middlesbrough at Old Trafford on Sunday.
Rostov: Medvedev; Bayramyan (Kireev 81), Mevlja, Navas, Kudryashov, Terentyev; Noboa (c), Prepelita (Devic 79), Erokhin; Azmoun (Bukharov 61), Poloz.
Subs not used: De Gea (GK), Young, Carrick, Lingard, Rashford.
Subs not used: Goshev (GK).