Astori's header from a Francesco Totti free-kick in the 17th minute hit the crossbar and the goal-line before bouncing out.
Replays showed the whole ball did not cross the line, but – after some discussion – the officials paid the goal and Roma went on to win 1-0.
Astori said officials made the right decision, with the result helping Roma move within a point of Serie A leaders Juventus.
"Perhaps my goal was controversial for some but for me there were no doubts," he said.
"I promptly celebrated the goal, there was a bit of hesitation by the officials over whether it crossed, but the referee had the guts to stand by his decision, which was right."
The 27-year-old added: "Yes, the officials had a discussion between them. I don't know who had the last word but they did what they are supposed to and allowed the goal."
Astori is already looking ahead to Sunday, when Roma host Lazio in the Derby della Capitale.
"This is a very special week leading up to the Roma derby. Both Lazio and ourselves arrive to the game in good form," he said.
"Today's win against Udinese was very important to help prepare for Sunday's derby and to put pressure on Juventus. However, we only think about ourselves and what we can do."
