Roma boss Eusebio Di Francesco said his side were waiting for Shakhtar Donetsk to "make a mistake" that would allow them to advance to the Champions League quarterfinals.
That mistake came in the 52nd minute as Edin Dzeko beat the offside trap to score the only goal of the game and push Roma through on away goals.
"My team played with focus tonight, perhaps without as much quality in the final third, but we knew that sooner or later they would make a mistake and we'd pounce on it," Di Francesco told Mediaset.
"We made a few too many mistakes in the first leg and they made the most of them. The thing I liked the most was seeing them in the locker room, as this was a victory achieved by men and a group that goes forward together.
"In terms of organisation, I think we are doing much better and we worked on a few set-play routines too. We should've cut out more of their passes to send the full-backs forward in the first half and perhaps their goalkeeper had more touches than anyone else, as they kept passing it back.
Shakhtar boss Paulo Fonseca saw the game a different way, as the Ukrainian side controlled 63 percent of possession, though they failed to put a shot on target while trying to hold onto their 2-1 first-leg advantage, and Ivan Ordets' 79th-minute dismissal made the task more challenging.
"I cannot judge [Roma] on one game, but I can say they made the most of their only opportunity," Fonseca said. "Shakhtar dominated the entire game and were always around the Roma area, even when going down to 10 men against 11.
"Roma didn't have many opportunities, we maintained possession and played in their half for most of it. We could've taken our chances better in the first leg, as if we had done that, we'd already be in the quarterfinals."
Goal scorer Dzeko had no doubts, saying Roma were "deservedly" through, and he said the chance to move on with Roma was a big reason why he stayed on in January, despite interest from Chelsea.
"I didn't want to leave, because all season we play for nights like this," he said. "We managed to do something Roma hadn't in 10 years, so going forward to play against the best makes us all very proud."
Roma will be returning to the quarterfinals of the Champions League for the first time since 2008, and Di Francesco said that no matter the result of the draw, "I want to keep on dreaming with this team."
"I was glad to see so many people in the stands cheering us on, suffering together and relishing every moment. The victory is experiencing nights like this," he added. "If we face any opponent with this attitude and determination, then we can go far."