The Serbian was encouraged by the side's outing against Crystal Palace, especially ahead of its trip to Olympiakos.
The 32-year-old skipper confirmed earlier this month that he will leave Old Trafford at the end of the current campaign, with Inter having all but finalized a deal to bring the Serbian to San Siro.
|MAN UNITED LATEST
"I believe we can do something special in the Champions League. I really do," the Serbia international told reporters after beating Crystal Palace 2-0.
"I believe that if we keep playing as we did — solid — and improve our form again we have a chance to do something good for the rest of the season. That means fourth place and the Champions League."
United ended a run of three games without victory on Saturday, with Robin van Persie and Wayne Rooney both netting at Selhurst Park.
Vidic believes there have been signs of improvement ahead of United's Champions League last-16 away tie against Olympiakos.
"We always believe we can do well, but we know we have to improve in every game and I think we saw some signs (at Palace)," he added. "To win it you have to defend well as a team. This is the way you should play and that is encouraging for us before we go to Greece."