The Reds led through Dirk Kuyt's first goal since December - but on-loan goalkeeper Luke Steele produced a string of oustanding saves on his debut to keep the Coca-Cola Championship side in the contest.
Benitez admitted his side's profligacy had proved costly yet again.
"We had a lot of chances but it was the same situation as in other games, we cannot take our chances to finish games. Today was the same," he told Sky Sports.
"I am really, really disappointed. We had chances, the keeper (Steele) making fantastic saves and they scored in the last minute.
"Sometimes this happens in football, it is the cup.
"It is difficult to explain, but it is the reality."
Howard scored as the Championship side came from behind to win 2-1 at Anfield in the fifth round.
Davey told Sky Sports News: "It's a fairytale, it's what dreams are made of - he's scored the winner in an FA Cup tie at the Kop end."
He added: "Credit to the players, they have been fantastic from the first minute and now they have got their rewards - they are in the next round.
"I always thought we would be in the game. We kept it tight, at half-time we were 1-0 down and if you get one chance and stick it away, you have got a chance. We kept it going and scored two great goals. I am pleased for them."
"It's come on his debut at Anfield in the fifth round. He pulled off some 'worldies'.
"Every player has stood up... it's a great day for the supporters and the board. Liverpool are a good side and put us under pressure. There were some fantastic last-ditch tackles in the box, and I thought it was a penalty at the end."
Match-winner Howard described the "best feeling" of his career. "We were hanging on defending, the 'keeper had an unbelievable debut then we nicked it at the end," he told Sky Sports News.
As for the goal, Howard said: "I thought I should have had a penalty at first - I was busy whingeing - then the ball fell to me and I thought about nothing but striking it.
"It's probably the best feeling I've ever had in football. I can't even describe it. I'm still away with the fairies at the moment.
"It was a real team effort today."
Steele, who only joined the club on loan this week, revealed he has barely familiarised himself with his new team-mates.
And he told Sky Sports: "I think it was Wednesday there was a call from West Brom saying Barnsley wanted me on loan and I accepted straight away.
"The lads defended brilliantly, obviously they were under a lot of pressure because Liverpool are such a good side.
"It's been a couple of months since I played last, so obviously I was just so happy to be out there and at Anfield. It's great I got a chance to play here."