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.
That gives Moyes' men, under pressure after a below-par season on the domestic front, a chance of pulling off a shock victory when they travel to the Allianz Arena for the second leg next week.
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 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.
Statistics showed United had only 22 per cent possession in a goalless first half but they probably squandered the best opportunity when Danny Welbeck failed to beat Manuel Neuer when clean through.
They upped their tempo in the second half and went ahead through captain Nemanja Vidic.
Bayern responded through Bastian Schweinsteiger but the German midfielder's night ended in frustration as he was sent off for a second bookable offence in the last minute.
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."
Bayern coach Pep Guardiola was unhappy about the dismissal of Schweinsteiger, claiming the decision by Spanish referee Carlos Velasco Carballo was "unfair".
Schweinsteiger was booked for a second time in the closing moments after a tussle with Rooney, and the player expressed frustration at the time by appearing to suggest he felt the Englishman dived.
Guardiola did not want to get drawn into a debate about the particular incident but did reveal he had spoken to Carballo about the matter.
The former Barcelona boss said: "I spoke with the referee and he gave me his opinion... I have to respect (it)."
When asked if he agreed with the decision, he said: "Absolutely not."
Asked about Schweinsteiger's apparent 'diving' gesture, he said: "I didn't see, but I spoke to the referee.
"He is a very good referee. He made a very, very good game, but it is unfair.
"But it is okay. To win the Champions League you have to solve and overcome everything. In football this kind of thing happens."
Guardiola, speaking in his press conference, also got involved in a spiky exchange with a journalist who asked him whether he thought United might have been negative.
"I didn't say that," said Guardiola, who was unhappy his inquisitor did not look him in the eye.
On the whole Guardiola was pleased with the performance, believing his side are well placed in the tie.
He said: "The result is 1-1 so we scored away, so this is good.
"I am happy for our performance, 70 per cent possession, 15 attempts.
"It is not easy to create the chances but we created quite enough.
"We were reacting good after going 1-0 down. It is not an exceptional result but it is good enough considering how good Manchester United are."
- Sports & Recreation
- Bastian Schweinsteiger
- Bayern Munich
- Pep Guardiola