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 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 pleased with the performance
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."