Webster holed his second shot to the par-five 10th at Doha Golf Club from 254 yards, recording the second albatross of his career with the same club and the second on the European Tour in the space of three weeks.
Holland's Joost Luiten made an albatross in the second round of the Volvo Golf Champions in Durban, coincidentally also on the 10th hole.
Webster added five birdies and one bogey to card a seven-under-par 65 and share second with South African Dawie van der Walt, whose compatriot George Coetzee claimed the lead with a superb 64.
"It was 254 yards to the pin, slightly into the wind off the left," Webster told Sky Sports. "I want to say five iron but it was a five wood and I hit a great shot, obviously straight at it.
"Because I carry a big crowd following me, there were about two people behind the green and one of them started jumping up and pointing down and I didn't know if it had gone over the back into the rocks or in! I got up there and he started clapping and it was a dream start."
Webster, who used the same five wood for an albatross in the Italian Open in 2011, added: "I have never been three under after one before and you wonder is this my day or have I had all my luck on the first hole, but I played great after that.
"I was really consistent, gave myself a lot of chances and made a few putts and was probably a little disappointed with seven under to be honest. I probably should have finished nine or 10 under but it's a good start."
Coetzee finished second here last year after Chris Wood snatched the title with an eagle on the 72nd hole and was fourth last week in Abu Dhabi, but the 27-year-old admitted he had enjoyed some good fortune.
"I think I'm quite lucky to get away with a couple of the shots I hit today," Coetzee told European Tour Radio. "It was another good putting display to shoot a nice number.
"I think I had 21 putts with a three-putt so that goes to show I can hit the ball a little bit better. I caught some good lies in the desert, I don't even hit it in the rough, just sideways and wide of the rough!
"I'm actually busy working on my swing and it's nice to shoot good numbers when you're working on stuff but I can't wait for things to click because then I might be unstoppable."
Van der Walt, who won the Nelson Mandela Championship in his native South Africa in December, carded a flawless 65 containing five birdies and an eagle as he looked to bounce back from a disappointing finish in Abu Dhabi.
The 30-year-old shot an opening 68 before fading to finish 71st and said: "I had a terrible week and looked like a blind squirrel, I couldn't do anything right. Although my game didn't feel too bad, so playing well today, I'm not too surprised.
"I'll just try and birdie every hole like I did today. I played solid and I forgot about last week, so just keep the same mind set and do the same things."
A 30-minute delay for morning fog meant three players were unable to complete their rounds, but 70 players were under par after the fog cleared to leave ideal conditions.
Former Open champions John Daly, Paul Lawrie and Ernie Els were all five under, Lawrie holing from 50ft for an eagle two on the 16th and Els holing out from a greenside bunker on the 18th for an eagle of his own.
European number one Henrik Stenson quickly returned to form after a rare missed cut in Abu Dhabi, the Swede carding birdies at his last two holes to complete a 68.
Playing partner Jason Dufner, the US PGA champion, returned a 70 but the third member of the marquee group, former world number one Luke Donald, could only manage a 72 which featured a missed par putt from inside a foot on the fourth.
Darren Clarke was forced to miss out on his 500th European Tour appearance after pulling out through injury. The 45-year-old suffered a pulled muscle on Tuesday and was forced to withdraw, with Sweden's Jens Dantorp - who earned his card through the qualifying school last year - taking his place in the field.
Clarke won the Open Championship in 2011 but has failed to record a single top-10 on the European Tour since his triumph at Royal St George's.