Crouch scored from a Fabio Aurelio cross in the 9th minute, and five minutes later midfielder Luis Garcia headed home from a Jermaine Pennant cross to send home fans into rapture.
The Galatasaray defence looked to be in disarray with the home team crossing at will and the Turkish side hardly able to get out of their own half.
The game looked to be wrapped up for Liverpool in the 52nd minute when Crouch met Steve Finnan’s cross with a stunning scissor-kick, crashing the ball past keeper Faryd Mondragon.
“Crouch’s goal was amazing. The movement was good and from this goal you can see he is a very good player,” Liverpool manager Rafa Benitez said after the match.
Galatasaray coach Erik Gerets made two substitutions at halftime, bringing on attacking players Hasan Sas and Umit Karan and changing to a 4-4-2 system in an effort to pull the match back.
The introduction of Karan proved to be a masterstroke with the substitute scoring two headed goals in quick succession in the 59th and 65th minutes to set up an enthralling final 25 minutes of the match.
Liverpool hung on however, with Gala striker Hakan Suker wasting two good chances late in the game to give the home side the full three points.
“That was a very difficult game, it could have been 4-0 but then they decided to play with two strikers and we couldn’t finish it,” Benitez told Sky Sports.
“I was worried at the end but it’s three more points.”
Hear no evil, speak no evil: (L to R)
The Galatasaray camp were ruing their missed chances, with coach Gerets highlighting his side’s slow start to the match and a lack of communication as their downfalls.
“At the start we forgot to play football because we were too impressed with the opponent,” said Gerets.
“The big difference between the teams is that in the Liverpool team everybody speaks but we don’t communicate enough – we could solve so many problems if we could speak to each other more.”
“We could have made a draw. We are crying, we are unhappy,” added Gerets.
Liverpool’s win took them to the top of Group C, equal on points with Dutch club PSV Eindhoven who also secured their first win with a 1-0 result at Girondins Bordeaux after a 65th minute strike from midfielder Mika Vayrynen.
Other Champions League results
Meanwhile in Group A, Chelsea took the outright lead with an easy 3-1 win over Levski Sofia in Bulgaria, with striker Didier Drogba scoring a hat-trick for the Londoners.
The Ivory Coast player scored in the 39th, 52nd and 68th minutes, with Levski substitute Mariyan Ognyanov scoring a consolation goal for the hosts late in the match.
On the ball: Didier Drogba scored
In the other Group A match Lionel Messi scored late for Barcelona to salvage a draw against Werder Bremen at the Weserstadion in Germany after a Carlos Puyol own goal had put the home side ahead after 56 minutes.
Group B action saw Bayern Munich go top with a 2-0 win over the struggling Inter Milan at the San Siro, handing Inter their second loss in as many matches in the Champions League.
Inter striker Zlatan Ibrahimovic received his second yellow card in the 58th minute reducing the hosts to 10 men, but it wasn’t until the 81st minute that Peruvian Claudio Pizarro opened the scoring for Bayern with a shot past Inter keeper Julio Cesar.
The Italian side went down to nine men in the 85th minute after World Cup hero Fabio Grosso was sent of for an elbow on Frenchman Willy Sagnol.
The numerical advantage allowed German striker Lukas Podolski to score his first Champions League goal in the 90th minute of the match to punctuate the victory for Werder.
First time in history
The other Group B match ended in a 1-1 draw after a Denis Boyarinstev goal for home side Spartak Moscow in the 5th minute was matched by a 59th minute strike by Luis Almeida da Cunha for Sporting Lisbon.
Vladimir Bystrov (R) and Rodrigo
Spartak and Sporting created history by playing the first ever Champions League match on artificial turf in their clash at the Luzhniki Stadium.
Finally in Group D, Spanish side Valencia took the lead in the group after a 2-1 win over Roma, while Ukraine’s Shakhtar Donetsk played out a thrilling 2-2 draw with Olympiakos from Greece in Kiev.
Shakhtar came from behind twice in the match to earn a point at home with goals to Francesco Matuzalem and Ciprian Marica cancelling out strikes from Michalis Konstantinou and Alberto Nery Castillo for Olympiakos.
At the Estadio Mestalla in Spain, Miguel Angulo and David Villa scored first half goals in Valencia’s win, sandwiching Roma player Francesco Totti’s 19th minute penalty.
The next round of UEFA Champions League matches will be played on October 17 and 18.