Giant Karlsruhe striker Josh Kennedy opened the scoring on 11 minutes, before Everton midfielder Tim Cahill and Palermo's Mark Bresciano gave the Australians a comfortable enough buffer to cruise through the second half.
"We played a fantastic first half. We did really well, we created chances, we scored great goals and we didn't give them a chance to play football," Verbeek said.
"It was exactly what the plans were before the game and I am very proud of what they did, especially in the first half, quality was very high and they began to get tired in the second half which we expected they would."
Iraq's 2010 World Cup qualifying campaign begins
Cheered on by 51,000 home fans, Australia could have won by more with only goalkeeper Mohammed Saqr preventing further damage in the first match of Asia's 'Group of Death,' which also contains Iraq and China.
The result vindicated Dutchman Verbeek's decision to select 10 overseas-based players in the starting eleven, with defender Craig Moore the only representative from the domestic A-League.
Moore was rock-solid at the centre of defence in his final international match and was substituted to a huge ovation with 12 minutes remaining.
Three first-half goals
The Socceroos began confidently and went ahead when Brett Emerton's cross from the right was headed in by German-based striker Kennedy.
The 1.94-metre forward posed a serious aerial threat in the Qatar box with Emerton and David Carney on the left feeding him crosses from both wings before he came off with am ankle injury in the second half.
The home side doubled their lead seven minutes later when Luke Wilkshire's corner was headed home by Cahill, who out-jumped naturalised Brazilian defender Marcone Amaral for his 13th international goal.
Australia were three-up in the 33rd minute when Bresciano sweetly swept home Celtic striker Scott McDonald's precision ball just inside the top of box.
Substitute Alhamad Mesaad had Qatar's best chance in the 58th minute but he sent a scuffed, side-footed shot to the outside of the left upright with the goal begging, while Yahya Ali Hassan had an effort saved by Schwarzer on the hour.
Japan beat 10-man Thailand
Meanwhile in Saitama, Japan overcame Thailand 4-1 in snowy conditions to win their opening Group 2 qualifier.
Midfielder Yasuhito Endo gave Japan the lead in the 21st minute with a superb free kick from 30 metres before Thailand stunned the home side by equalising within a minute when Teerathep Winothai smashed a fierce shot past Japan keeper Yoshikatsu Kawaguchi from the edge of the box.
Japan were booed off at halftime by the crowd of 35,000, but went back in front 10 minutes after the interval when striker Yoshito Okubo slotted home from close range.
Things got worse for Thailand when Narongchai Vachiraban was sent off before Japan defender Yuji Nakazawa made the points safe by heading the home side's third in the 66th minute, and substitute Seiichiro Maki added a fourth in injury time.
Later in Seoul, Korea Republic scored a 4-0 win over Turkmenistan with Kwak Tae-Hwi opening the scoring, Seol Ki-Hyeon bagging a double, and Park Ji-Sung also netting for the home side in Group 3.
China draw with 10-man Iraq
In Dubai, China came from behind to claim a 1-1 draw against 10-man Iraq in the other Group 1 match, with playmaker Nashat Akram sent off for a second bookable offence in the 71st minute.
After a goalless first half of a spiteful match, Iraq were awarded a 50th minute penalty which midfielder Hawar Taher audaciously chipped over the China keeper to make it 1-0.
However with hard and late tackles flying in from both teams, it was only a matter of time before a player was dismissed, and Akram, who recently missed out on a move to Manchester City due to work permit issues, was the man who received his marching orders.
China took immediate advantage, with Zheng Zhi heading home powerfully from a corner to level the scores with 15 minutes to play, and although China had the best chance to win, the match ended even.
Earlier in Tehran, home side Iran were held to a 0-0 draw by Syria in Group 5.