The Imps had beaten the likes of Ipswich Town, Brighton & Hove Albion and Burnley to become the first non-league side to reach the last eight since 1914, and they held their own for much of the opening 45 minutes against Arsenal today.
"We had a tough second half against world class players, when they scored on the edge of half time it was always going to be difficult," Cowley told BT Sport.
"They got confident from then on in and we got dragged around and hurt in wide areas. But we're the first non-league team in over 100 years to get to the last eight of the FA Cup. We need to be proud of what we've done.
"This is not a time for criticism, it's a time to reflect on what we've done. The key thing is to learn from this and come away better footballers and better people."
Lincoln could still reach Wembley for the first time in their history with a win over York City in their two-legged FA Trophy semi-final next week.