A goal kick from Roma's Morgan De Sanctis landed just inside the Verona half before Destro controlled the ball, saw Verona goalkeeper Pierliugi Gollini off his line and scored with a high, dipping shot from 45 metres.
Roma maintained their 100 percent record after five Serie A games as did Juventus who won 3-0 at Atalanta, helped by two Carlos Tevez goals and a penalty save from Gianluigi Buffon which meant that the champions have still to concede a goal this season.
Roma visit Juventus in their next game on Oct. 5.
Totti, in his 23rd season at his only professional club, twice came close to scoring, and substitute Alessandro Florenzi, who replaced the Roma captain just after the hour, broke the deadlock for the hosts at the Stadio Olimpico.
As usual, Totti, playing just behind the front two, was involved in nearly all Roma's attacking moves.
He thought he had given his side the lead when the ball broke to him in the penalty area with Gollini stranded, but Frederik Sorensen cleared the ball off the line and it hit the post on its way to safety.
Totti was close again with a side-footed volley from Maicon's pass early in the second half. This time, the 19-year-old Gollini turned the ball over.
Totti also had time to nutmeg an opponent before being taken off, as expected, just after the hour to a standing ovation.
Gervinho came on in the second half and his entrance gave fresh impetus to Roma, who until then had struggled to break down a well-marshalled Verona defence.
Florenzi broke through with 15 minutes left when he met a loose ball with a first time shot from 25 metres. Destro missed an easy chance after a mazy Gervinho run but atoned with his incredible strike shortly afterwards.
"Adem Ljajic and the captain (Totti) played really well. They tired out Verona and this allowed their team mates to settle the match at the end," Roma coach Rudi Garcia told reporters.
"Destro's goal showed great class, the instinct of a hitman, the work of a pure attacker.
Destro added: "I've been thinking of a goal like that for a long time, catching the goalkeeper out of his goal. I tried it today and it went well.
Argentine forward Tevez gave Juventus a 36th-minute lead when he poked home a loose ball from close range.
The game swung decisively in Juve's direction in 60 remarkable seconds just before the hour.
Atalanta were awarded a penalty after a clumsy challenge by Giorgio Chiellini but, after a stuttering run-up, German Denis saw his weak effort saved by Buffon. Juve went straight down the other end and Tevez scored with a low shot inside the post from the edge of the area.
Alvaro Morata wrapped up the win by heading his first goal for Juve since his move from Real Madrid.
(Writing by Brian Homewood in Berne; Editing by Ian Chadband)
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