Colombian Morelos received two yellow cards within seconds of each other for kicking the ball away and dissent eight minutes before the break, while Flanagan, booked in the first half, was then dismissed after an aerial challenge left his opponent crumpled in a heap.
The Russians launched wave after wave of attacks on the Rangers goal but could not breach their defence and the Scottish side clung on until the final whistle.
"We apologise to the fans and to our team mates -- they have done incredibly well and have got us out of jail," Flanagan, who joined the club on a free transfer in the close season, told RangersTV.
"I'm personally disappointed at letting my team mates down, but the main aim here was to come and get to the group stages."
Rangers, leading 1-0 from the first leg, got off to the best possible start at the Stadion Neftyanik when Ovie Ojaria, on loan from Gerrard's former club Liverpool, opened the scoring in the ninth minute.
Ufa equalised in the 32nd minute when Dmitry Sysuev sprung the offside trap to beat goalkeeper Allan McGregor and while the Russian Premier League side were camped out in the Rangers half for the majority of the second half they could not find the two goals they needed to advance.
Gerrard, who took over at Rangers in May and has brought in 12 players as he looks to end the dominance of Glasgow rivals Celtic, said it was a "big moment" for the club, who are still recovering from financial problems that saw them reformed in the bottom tier of Scottish soccer in 2012.
"I think it's what we deserve and tonight we became a team and we've relaunched Rangers because I feel like we have achieved something together," he said after the match.
Celtic will join Rangers in the group stage of the Europa League after they cruised to a 3-0 win over Suduva on Thursday to win the tie 4-1 on aggregate.
(Writing by Peter Rutherford in Seoul; Editing by Greg Stutchbury)
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