Astori's header from a Francesco Totti free kick bounced down off the underside of the crossbar and into the arms of Udinese goalkeeper Orestis Karnezas in the 17th minute.
The goal-line official signalled that it was not a goal but was then over-ruled by referee Marco Guida, according to both Astori and Udinese coach Andrea Stramaccioni.
Goal-line technology is not used in Serie A, although the Italian league employs additional assistant referees, a system pioneered by UEFA featuring an extra official on each goal line.
Television replays were inconclusive because of a goal post obstructing the view of the ball. However, still photographs taken from a different angle appeared to show that the ball had crossed the line by several centimetres.
"The goal was legitimate and the referee did well to spot it, but I haven't changed my mind, I'm in favour of technology as it would give some extra help to the referees," Roma coach Rudi Garcia told reporters.
Astori said the referee had over-ruled his assistant.
"The assistant said it was no goal, but the referee had the courage to give the goal, he took the responsibility and he was right" he said.
"I have to hand him my compliments. It's difficult to judge on the pitch, in a split second."
Stramaccioni, however, said the incident made a mockery of having goal-line officials.
"The goal-line officials were brought in to decide on the goal/no-goal decisions," he said. "It's a structural error: If you're not going to respect their judgement, there's not much point in having them there."
There was more controversy in the second half when Panagiotis Kone appeared to be bundled over by Urby Emanuelson but the home team's appeals for a penalty were turned down.
The win drew Roma level on 39 points with Serie A leaders Juventus, who host Inter Milan later on Tuesday (8:00 p.m.).
(Writing by Brian Homewood in Berne, editing by Ed Osmond)
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