Fleischmann, 26, has four goals and six assists in 23 games this season. Last year, he had offensive career-highs with 23 goals and 28 assists.
The Czech Republic-native had been the subject of trade rumours for several weeks. Some reports suggested that the Los Angeles Kings also had interest in Fleischmann.
Fleischmann started the season as Washington's number-two centre, but he has recently been a healthy scratch for Washington.
"In Tomas we are acquiring a top-six forward who is only 26 years old," said Avalanche GM Greg Sherman in a press release. "His addition will bolster our lineup immediately."
Hannan, 31, waived his no-trade clause in order to be dealt to the Capitals. A veteran of 11 NHL seasons, Hannan will be joining his third club after playing with the Avalanche and the San Jose Sharks.
The native of Richmond, B.C. has six assists in 23 games this season.
Capitals general manager George McPhee has been in the market for a shut-down defenceman for some time. Hannan's experience is expected to provide more consistency to Washington's blueline, as the Capitals have been mostly playing the duo of Karl Alzner and John Carlson against opponents' top lines.