The Tigers face the Llanelli Scarlets next Saturday in the Heineken Cup semi-final and are guaranteed a place in the Guinness Premiership play-offs.
Howard said: "I am extremely confident. What we have set ourselves as a target is not an easy target. A lot of factors will have to go with us.
"We aim high, we believe in ourselves, the team has been winning and we are playing well.
"This is a nice start and having the monkey off your back doesn't hurt."
Leicester scored through Tom Varndell, Tom Croft, Ben Kay and Alesana Tuilagi to place one hand on the trophy at half-time.
But the Ospreys hit back through Lee Byrne and closed to within three points at one stage after Shane Williams had scored twice.
Varndell sprinted in for his second and although Nikki Walker responded for the Ospreys, Leicester were able to hold out to clinch the trophy.
Howard said: "For the neutral it was a fantastic game. For us it was a bit nailbiting at the end. I am extremely happy with the victory.
"The win was extremely important. I think we played really well at times."
Leicester now face a six-day turnaround before tackling the Scarlets and Howard added: "We have to manage the squad the best we can."
Ospreys coach Lyn Jones was left to rue his side's first-half performance.
"That hurt us," he said. "We turned the ball over five times in first 30 minutes which is unusual of us. We did not function in that first half. We committed far too many mistakes to be efficient at this standard.
"When you are shelling four tries to sides like Leicester it is difficult to fight back."
Jones insisted he never lost faith that the Ospreys could win, even when they went into half-time 21 points down.
"Once we scored first after break the supporters woke up and gave us a cheer. We dragged back to within three points and felt it was coming our way.
"I am really proud of my players for a massive effort."