The hosts began brightly and almost took an early lead against their Russian visitors when Daniel Candeias's volley was touched over the crossbar by goalkeeper Aleksander Belenov.
Gers continued to knock on the door as the game went on as Morelos and James Tavernier both weighed in with efforts on target, only for the Ufa stopper to make straightforward saves.
Eventually the deadlock was broken shortly before the break, with Ryan Kent's corner headed goalwards by Morelos as Belenov parried the effort into the path of Goldson, who duly rifled home.
The second half proved less eventful as Rangers continued to probe for a second to little avail, with Kent firing just wide on 71 minutes and a penalty appeal by Glenn Middleton turned down.
Steven Gerrard's side would fail to add to their tally on the night but will nonetheless take a slender lead to Russia next Thursday in the return leg.
RANGERS (4-2-3-1): McGregor; Tavernier, Goldson, Katic, Flanagan; Jack, Ejaria; Kent (Middleton 77'), Arfield, Candeias (Lafferty 69'); Morelos
FC UFA (4-3-3): Belenov; Zhivoglyadov, Tabidze, Nedelcearu, Alikin; Salatic, Zaseev (Vanek 85'), Jokic; Paurevic, Igboun (Krotov 90+3'), Oblyakov (Carp 83')