The triple European champions failed to progress from the group stage in 2011/12 after suffering a shock 2-1 defeat by Swiss club Basel in their final fixture.
"We've lost two finals to Barcelona and won two other finals in the last decade or so but we want to do better," Ferguson told the club's website. "Certainly we'll do better than last year - no question about that.
"The European Cup is incredible. There's an atmosphere in the build-up to a European final and great charisma about it.
"Also, without question, it elevates your status within the game like the Real Madrids and AC Milans. We want to be up there in terms of the number of trophies won in European football."
Starting with their defeat by Barcelona in the 2011 final, United have won just three European matches out of 11 and have lost four of their last five.
Ferguson's side, playing in the competition for a record 18th time, will be without long-term injury victim Wayne Rooney on Wednesday but on the plus side they have a red-hot Robin van Persie ready to make his first European start for the club.
Van Persie has been in good form since leaving arch-rivals Arsenal in the close season and is already the joint top scorer in the Premier League with four goals, alongside Swansea City's Michu.
The Dutch international was left out of the starting line-up on Saturday before coming on as a second-half substitute to help United crush Wigan Athletic 4-0 in the league.
Fellow substitute Nick Powell capped his United debut with a superb long-range goal against Wigan and the former Crewe Alexandra player said he had some nervy moments when he entered the fray in front of a 75,142 crowd at Old Trafford.
"When I first came on it scared me a little bit because I'm only used to 5,000 people now and then," said the 18-year-old midfielder. "But it's a great atmosphere and the fans were great.
"I learn every day especially from the midfielders Paul Scholes, Michael Carrick, Ryan Giggs. They've all been through it and won trophies and that's what I want to do."
Galatasaray are making their first appearance in the Champions League for six years.
The Turkish club have not won any of their seven away games against English teams but can count creditable draws against Manchester United, Liverpool and Leeds United in their record.
Galatasaray are leading their domestic league, having won three games out of four this season, but coach Fatih Terim knows they are going into the match at Old Trafford as underdogs.
"We will be playing against one of the best clubs in the world," he said. "Can anyone deny they are more likely to win?
"We will try our best, we will never give in. We may lose but we're not afraid to lose. My players and I are experienced enough to know we will have to put up a better fight there than we do in the domestic league."
Galatasaray are without injured central defender Tomas Ujfalusi and right back Sabri Sarioglu.
Umut Bulut, who has scored five goals in four league games this season, is set to partner former Bolton Wanderers forward Johan Elmander in att
