The hosts dominated the fifth-round tie after falling behind, Joshua King grabbing a hat-trick against a visiting side who were reduced to 10 men following Geoff Cameron's dismissal on the stroke of half-time.
And Bowyer heaped praise on his squad, insisting that the convincing win was no more than they deserved.
"For myself and the players I think it is probably our biggest achievement in the time that I have been in charge," he said.
"The lads are buzzing, they are delighted and rightly so – it was a real Cup tie.
"Full praise to them and full credit to them, they deserve every plaudit going.
"Against a quality team the lads worked ever so hard. It is a great example for them in what you can achieve when you do that."
Bizarre Jose Mourinho statement aims to pre-empt punishment after Fenerbahce boss clashes with Galatasaray counterpart
'Andy Robertson has a decision to make as a player, where you either roll your sleeves up and give someone great competition or you accept decisions and say that your time is done.' Alan Shearer on the conundrum facing Arne Slot's Liverpool
