DOHA, January 22, 2014 (AFP) - Englishman Webster, who started out on the par five 10th tee, saw his five-wood approach from 254 yards disappear into the cup.
But despite that feat he eventually had to settle for a share of second place with Dawie Van der Walt after Coetzee finished with three consecutive birdies to take a one-shot lead with an eight-under 64.
Van der Walt had earlier carded a fluent 65 to take the provisional clubhouse lead but Webster was set to overtake him after being four under after just three holes.
But after a steady display, a bogey on his 15th saw Webster end up in a tie for second with Van der Walt who had carded a bogey-free round.
Webster's albatross was the second of the season on the European Tour after Dutchman Joost Luiten produced one at the Volvo Golf Champions in Durban earlier this month.
However, this was the first time at this level that anyone had shot an albatross on his opening hole.
Webster was pleased with what he called a "dream start".
"254 to the pin, wind slightly off the left. I was going to say 5-iron but it was a 5-wood. Hit a great shot obviously straight at it," said Webster.
"Because I carry a big crowd following me, there was about two people behind the green and one of them started jumping up and pointing down. Thought, don't know whether it's gone over the back to the rocks, get up there and he starts clapping. Dream start.
"I played great after that and was just really consistent, gave myself a lot of putts and gave myself a lot of chances. I was probably a little disappointed with 7-under to be honest. I should have finished probably nine or ten, but it's a good start."
Coetzee, who was tied second with Sergio Garcia at last year's Qatar Masters won by Chris Wood, played a brilliant back nine which he began with an eagle on the 548-yard 10th, where Webster had recorded his feat.
However, he dropped a shot on the 13th before three straight birdies from the 16th helped him clinch the lead.
Spaniards Rafa Cabrera-Bello and Elviro Nacho shared fourth on 66 along with Frenchman Francois Calmels and England's Ross Fishe.
A group of six players including American John Daly, South African great Ernie Els and India's Gaganjeet Bhullar, were a further shot down on 67.
Defending champion Wood had a forgettable first round as he shot a 73 to be in a tie for 85th position.
Van der Walt said he will try to keep his focus in the latter rounds, drawing lessons from last week's Abu Dhabi Golf Championship where he carded an impressive 68 in the first round before fading away to finish a poor 71st.
"I had a terrible week and looked like a blind squirrel, I couldn't do anything right," said the South African.
Nacho, too, had a bogey-free round, while Frenchman Calmels dropped two shots that somewhat blotted an otherwise spectacular round in which he had six birdies and an eagle.
The 2006 champion and European number one Henrik Stenson hit form with a 68 after missing the cut last week in Abu Dhabi while his playing partner Jason Dufner of the US carded a 70.
Irishman Darren Clarke was forced to miss out on his 500th European Tour appearance after pulling out with a muscle injury.
Leading first round scores at the Qatar Masters in Doha on Wednesday (Gbr & Irl unless stated, par 72):
64 - George Coetzee (RSA)
65 - Steve Webster, Dawie Van Der Walt (RSA)
66 - Ross Fisher, Rafael Cabrera Bello (ESP), Francois Calmels (FRA), Nacho Elvira (ESP)
67 - John Daly (USA), Thomas Aiken (RSA), Ernie Els (RSA), Paul Lawrie, Branden Grace (RSA), Gaganjeet Bhullar (IND)
68 - Shiv Kapur (IND), Mikko Ilonen (FIN), Tom Lewis, Matthew Baldwin, Simon Dyson, Seve Benson, James Morrison, Thongchai Jaidee (THA), Thorbjorn Olesen (DEN), Alejandro Canizares (ESP), Henrik Stenson (SWE), Michael Hoey
69 - Richard Bland, John Parry, Magnus A Carlsson (SWE), Kristoffer Broberg (SWE), Marcel Siem (GER), Darren Fichardt (RSA), Johan Carlsson (SWE), Alvaro Quiros (ESP), Tyrrell Hatton, Jose-Maria Olazabal (ESP), Peter Hanson (SWE), Stephen Gallacher, Fabrizio Zanotti (PAR), Eddie Pepperell, Christopher Doak
70 - Robert Jan Derksen (Ned), Marco Crespi (ITA), Soren Kjeldsen (DEN), Damien McGrane, Jason Dufner (USA), Martin Kaymer (GER), Adrien Saddier (FRA), Brooks Koepka (USA), Robert Karlsson (SWE), Anthony Wall, Robert Rock, Peter Uihlein (USA), Marc Warren, Dominic Foos (GER), Simon Thornton, Gregory Havret (FRA)
Selected
71 - Jeev Milkha Singh (IND), Hennie Otto (RSA), Pablo Larrazabal (ESP), Miguel Angel Jimenez (ESP), Sergio Garcia (ESP), Justin Walters (RSA)
72 - Garth Mulroy (RSA), Jbe Kruger (RSA), Wade Ormsby (AUS), Luke Donald
73 - Ali Al-shahrani (QAT), Jin Jeong (KOR), Brett Rumford (AUS), Paul McGinley
74 - John Hahn (USA), Richard Green (AUS), Edoardo Molinari (ITA), Marcus Fraser (AUS), Richard Sterne (RSA)
76 - Connor Arendell (USA), Shane Lowry, Brandon Stone (RSA)
78 - Ali saleh Al kaabi (QAT)