tee
The right tool for the right task: 'tee'
tee - read from standard input and write to standard output and files
So your commandline becomes
% echo "output" | sudo tee -a file
The advantage of tee over executing Bash with administrative permissions is:
You do not execute Bash with administrative permissions
Only the 'write to file' part runs with advanced permissions
Quoting of a complex command is much easier