The Giallorssi won the game courtesy of a first goal for Argentine defender Walter Samuel who connected with a far post header after almost 70 minutes of the game.
Roma were anything but impressive however, and could have lost the game had Lecce been awarded what appeared to be two perfectly legitimate penalties.
Delvecchio, who did put the ball in the net with aplomb but had it wiped out for a push on a defender, admitted that things must improve.
"It was tough against Lecce because they defended well," said the Italian international. "It was a vital win because both Lazio and Juve won their games.
"I think it is fair to say that against Liverpool we need to find the other Roma, the one that has put us top of the league."
President Franco Sensi stated the importance of coming from behind to progress past Liverpool, who go into the second leg with a two goal cushion.
"We have to go through, we are in the hands of the ball, the players and god."
Lecce could easily have snatched a share of the points in Sunday's game, but found a referee seemingly not prepared to risk the wrath of the Stadio Olimpico by awarding a penalty against the home side.
The first appeal involved a clear handball by Carlos Antonio Zago that saw the referee actually put his whistle to his mouth before changing his mind - and then booking Cristiano Lucarelli for his excessive appeals.
Zago denied that the handball had been deliberate, stating: "The ball hit my arm, but I didn't want to touch it.
"I told Lucarelli that these things sometimes happen in football, and that he should not be mad about it."
Sensi also believed that both incidents were clear-cut, adding: "The rules should be revised. When Zago stopped the ball with his hand it was a penalty, as it was a penalty when Lecce's Dainelli stopped the ball with his hands, and when Samuel brought down Lucarelli."
Lecce boss Alberto Cavasin was magnanimous in defeat, and refused to blame the official, saying: "I cannot judge the referee because it is not my job. I think he did a good job. I would prefer to give my opinions on the mistakes that we made today, although we did play very well.
"I will never work at Roma because I always seem to lose here so they will want to keep my as the opposition."
Finally, Fabio Capello was happy to have escaped with three points, and that his team had shown spirit when things were not going their way.
"Some games you win by playing well, but sometimes you can win by showing your fighting spirit." Capello said.
"Lecce drew in Parma and were tough for Lazio to beat, so I am satisfied with three points."
