The Imps beat Premier League team Burnley, who are 81 places above them in the tables.
Pundits have described the 1-0 victory as one of the biggest upsets in football history.
Sean Raggett scored an 89th-minute winner by squeezing a header over the line following a late corner at Turf Moor.
The Lincoln manager said he was "lost for words" after the final whistle.
"Last eight of the FA Cup - it sounds really good," Cowley said. "I'm lost for words."
Lincoln are the first non-league team to reach the FA Cup's last eight since Queens Park Rangers in 1914.
They are the eighth side from below England's four fully professional divisions to have beaten top-flight opposition since World War II.
They are only the second to have done so since 1989.
Lincoln's run to this round has included victories over Championship sides Ipswich and Brighton - but this is by far their biggest scalp.
The feat is especially impressive given that top-flight Burnley have won seven of their previous eight home matches in all competitions.