Eleven days after drawing 0-0 at the Abbey Stadium, United were far more convincing against their League Two opponents at Old Trafford.
Juan Mata, Marcos Rojo and James Wilson all found the net in the fourth round replay at Old Trafford on Tuesday night.
With Chelsea and Manchester City out of the FA Cup, United are the highest-ranked team left in the competition and they will face League One side Preston in the last 16.
Van Gaal's message to the fans after the final whistle was clear enough - he fancies his chances of going all the way in a competition United have not won in 11 years.
"We have a real chance now," the United boss said.
"Why? Because we are the highest-qualified team in the FA Cup."
"I am not a casino man," the Dutchman added.
"But of course I think the highest-qualified teams have more possibilities to win than lower-qualified teams
"Maybe people are maybe betting on us
I am not allowed to (but) I give you a tip."
Van Gaal has enjoyed success in cup competitions throughout his career.
The 63-year-old lifted the Copa del Rey with Barcelona and also won the KNVB Cup with Ajax and the DFB Pokal with Bayern Munich.
And with United trailing leaders Chelsea by 10 points in the Premier League, the FA Cup seems the best chance the Dutchman has of winning silverware in his first year at Old Trafford.
"For the club, the fans, the coach and my staff, we want to win a title this year and the biggest chance is the FA Cup," Van Gaal said.
"For me, personally, I won always a title in my first year everywhere I worked."
Van Gaal found life difficult in the original fixture at Cambridge just under two weeks ago, and it was the same story for the first 25 minutes of the replay.
In fact, the hosts should have gone behind in the first minute.
Just 50 seconds had elapsed when Daley Blind accidentally knocked the ball into Tom Elliott's path, but luckily for United, his curling shot hit the post and bounced out to safety.
Mata finally breached the Cambridge defence when he stabbed in Marouane Fellaini's knock down in the 25th minute and soon after Rojo headed in Robin van Persie's cross to clinch his first goal for the Red Devils.
Second-half substitute Wilson rounded off the win with his second goal of the season, and Van Gaal was happy to see his team avoid the same fate as Chelsea and City, who were eliminated by Bradford and Middlesbrough respectively.
"I am very happy," Van Gaal said.
"If they had scored the first goal then it would be a very difficult game.
"But I am very pleased because it was 3-0 and we have a clean sheet and we are in the next round."
The only slight disappointment for Van Gaal was the performance of Robin van Persie, who came off in the 66th minute and went straight down the tunnel.
"Robin van Persie had four possibilities to score," Van Gaal added.
"But I am pleased because you have to beat your opponent and then you are in the next round.
"I took him off because I want him to play against West Ham (on Sunday)."
Cambridge's fairytale run through to the fourth round may have come to an end, but their manager Richard Money enjoyed the trip to Old Trafford, which is set to net the club up to £1.5million.
"It means we can be safe and stable financially," Money said.
"We are not going to have a spending spree in the summer and have one massive effort to be promoted.
"We need to grow
We need a new stadium and a new training ground or new facilities at the training ground
That is what the cup run will do for us."
Source : PA
Source: PA