Brighton took the lead through Richie Towell but found themselves on the receiving end of a giant-killing as Lincoln reached the competition's last 16 for the first time in over a century.
Cowley told BBC Sport after the match: "You always go into games thinking you have a chance. The boys were organised and motivated.
"I was really pleased that we were able to stay in the game [after going behind]. Half-time gave us a chance to regroup. We were brave all afternoon, we wanted to press high and be committed to our style and what we believe in.
"I can't believe the scenes and the emotion and the support – it's beyond all of my wildest dreams."
The result saw Brighton eliminated from the FA Cup by non-league opposition for the first time since 1998, when they lost to Hereford United.