The last-16 first leg loss at home came after three games without a win in Serie A while the likely sale of the indebted club to an American consortium has added to the uncertainty surrounding under-fire coach Claudio Ranieri's future.
Ranieri has vowed not to quit and Riise, whose mistake let in Luiz Adriano for the Ukrainians' third on their knockout stage debut, tried to take the blame as Serie A was left reeling again following AC Milan's loss to Tottenham on Tuesday.
"I feel bad, my error was incredible. The fans are angry with me and I say sorry to everyone, team mates included. I don't think we played badly," Norwegian left back Riise, substituted at the break following the mistake, told Sky.
"I was ready to come back on but Ranieri told me I was coming off, I think for the error. I hope the fans know that I always give everything."
Roma took the lead in the 28th minute through midfielder Simone Perrotta's header but Shakhtar equalised a minute later through Jadson's fortunate deflection before fellow Brazilian Douglas Costa netted on 36 minutes.
Following Riise's aberration, Jeremy Menez picked his spot in the top corner after the break to give Roma some hope for the second leg in Ukraine on March 8.
Ranieri was upbeat about the performance and with a 2-0 win in Donetsk being enough to send the Romans through, the doom-mongers are perhaps a little premature despite Roma's eighth place in the Italian league.
New owners will also add impetus, whatever Ranieri's future, and a club the size of Roma can do little more than aspire to reach the Champions League last eight having surprised every fan in Italy with their stunning revival last season.
Whether Riise will be present in Donetsk remains to be seen.
