Yes, you are allowed to buy and use Power cards immediately after eliminating a Monster, as long as it is still your turn and the game has not ended.

Explanation

• According to the turn structure on page 4, the steps are:

  1. Roll Dice
  2. Resolve Dice
  3. Enter Tokyo
  4. Buy Power Cards
  5. End of Turn

• The elimination of a Monster can occur during the Resolve Dice step or as a result of a Power card effect. The Buy Power Cards step comes after this, so you may still buy Power cards after eliminating a Monster, provided the game has not ended.

• The game ends at the end of a turn when a Monster reaches 20 Victory Points or if only one Monster remains (page 6). If you eliminate a Monster but there are still at least two Monsters left, the game continues and you proceed to the Buy Power Cards step.

• If, during your turn, you eliminate all other Monsters and are the last one remaining, the game ends at the end of your turn (page 6). You may still complete your turn, including buying Power cards, before the game officially ends.

Source: pages 4, 5, and 6.

Source: Core Rulebook (pp. 2, 3, 4, 5, 6)