United were expected to struggle as they hosted the European champions at Old Trafford but soaked up a lot of pressure and earned a creditable 1-1 draw in their quarter-final first leg.
That gives Moyes' men a chance of pulling off an unexpected victory when they travel to the Allianz Arena for the second leg next week.
"I always felt that we could score a goal against them, and I'll go to Munich believing that we can score a goal," Moyes told Sky Sports News.
"Coming into the tie, most people had Bayern as favourites for the game and rightly so – they are European champions.
"But I think this club is used to big nights in Europe and I think tonight it showed it on the field and off the field as well."
Asked if it was important Bayern knew they faced a battle in the second leg, Moyes said: "I believe they had a fight on their hands before the first leg as well.
"Hopefully tonight shows we will go there, and we will go toe to toe with them and we will try to come out on top. It is a tough tie but but we all believe we have got a chance.
"Our best football has been in this competition and we will need to show our best football when we go to Bayern next week.
"We need to score. We need to go there and attempt to win the game and get some goals.
"Other sides have gone there done that and we will need to do the same. Tonight's performance was more what you'd hope for."
Moyes had mixed feelings at the end of the night, having been pleased with the performance but disappointed his side could not snatch a win.
They squandered the best opportunity of the first half when Danny Welbeck failed to beat Manuel Neuer when clean through.
Moyes said: "I am happy with the performance, but a bit disappointed with the result in the end.
"I thought the players put in a great effort, great commitment tonight.
"I'm a bit disappointed we couldn't take one of the other chances we had but we played a great team tonight in Bayern Munich. I'm sure everyone saw that."
Moyes admitted Welbeck, having been played through by Wayne Rooney, should have done better.
He said: "It was a great chance. It was very good goalkeeping as well, he stood up, the goalkeeper, and he missed it but it was a great opportunity for us at that time of the game."
Skipper and goalscorer Nemanja Vidic is also confident United can get a result in Bavaria.
"It was a tough game and a good result - and I think it will be the same game next week," Vidic said.
"They will have the same possession, especially as it's their own ground, but if we keep the shape we can hurt them with our pace.
"We felt we had good positions and were solid, but one mistake was punished and they showed what a good team they are.
"They had a lot of possession but we accepted they would and we tried to counter attack."