The Giallorossi ran out 2-1 winners over Trabzonspor in the first leg of their Europa Conference League play-off tie, with Shomurodov's close-range finish erasing Andreas Cornelius's earlier equaliser for the hosts.
The result means avoiding defeat in a week's time will be enough for the Giallorossi to reach the group stages of the European competition, which is in its first season.
With a raucous crowd inside the Senol Gunes Arena formally welcoming the Giallorossi back to action, Mourinho's side started off somewhat quietly - with clear-cut chances few and far between in a first half of little real action.
What danger the visitors were creating came from the flanks, although often the final cross was marginally overhit. That changed early in the second half, however, as Henrikh Mkhitaryan measured his delivery perfectly for Lorenzo Pellegrini to steer in at the back post.
That seemed to settle Roma - at least until Cornelius announced his arrival off the bench with an almost-immediate goal that delighted the home faithful.
That header immediately threatened to switch the momentum, with Trabzonspor forward Anthony Nwakaeme perhaps guilty of wasting a glorious chance to put his side ahead.
And he was soon punished for that, as Shomurodov reacted quickest after Gianluca Mancini's header bounced back off the crossbar to steer it in and give Roma the lead again.
From that point on it was about seeing the contest out; something Mourinho's side did comfortably.
With the second leg taking place at the Stadio Olimpico next Thursday, the Giallorossi have the small matter of their first Serie A game to worry about first: they host Fiorentina on home turf on Sunday night.
TRABZONSPOR 1-2 ROMA
SCORERS: Cornelius (64'); Pellegrini (55'), Shomurodov (81')
TRABZONSPOR (4-3-3): Cakir; Peres, Edgar Le, Vitor Hugo, Koybasi; Ismail (Trondsen), Berat (Omur), Hamsik; Gervinho (Cornelius), Djaniny (Koita), Nwakaeme
ROMA (4-2-3-1): Rui Patricio; Karsdorp, Mancini, Ibanez, Vina; Cristante, Veretout (Kumbulla); Zaniolo (Reynolds), Pellegrini (Diawara), Mkhitaryan (Perez); Shomurodov (Mayoral)
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