A try had seemed unlikely in a match dominated by kicking on Friday night, with Worcester replacement Joe Carlisle seemingly having won the game for the Warriors with three second-half penalties.
However, Birkett crossed as the clock ticked down to steal the victory for the London visitors and move them up to fourth in the table.
Worcester were ahead after four minutes when Andy Goode kicked an angled 30-metre penalty, but it was Wasps who had the better of the opening exchanges as they dominated in terms of possession and territory.
Careless mistakes from Steve Thompson and Joe Simpson hindered their progress, but eventually their efforts were rewarded when Nicky Robinson tied up the scores with a 45-metre penalty.
Wasps continued to maintain their stranglehold on the game with clever kicking from Robinson ensuring that the Warriors were playing on the back foot.
The visitors then had a chance to take the lead when Marcel Garvey was harshly adjudged to have impeded Simpson, but Robinson's 40-metre kick went hopelessly wide.
The game continued to be a poor one with both sides content to kick for possession. Errie Claasens and Miles Benjamin did try and put a bit of sparkle into the proceedings, but their enterprising runs came to nothing for Wasps.
Robinson was narrowly short with another penalty attempt and then, on the stroke of half-time, the line was threatened for the first time in the match.
Worcester lost possession in their own 22 and Riki Flutey and Simpson both made half breaks, although the Warriors kept their line intact.
However, Robinson did give his side a deserved 6-3 interval lead with a neat drop-goal with the last kick of the half.
Goode was replaced at the interval and his replacement Carlisle put his side into the opposition 22 with his first kick.
This set up Worcester for their first period of pressure and they drew level when Carlisle kicked a 30-metre penalty after 46 minutes.
Wasps regained the lead four minutes later when Robinson kicked a 45-metre penalty after the home side were penalised for a scrummage offence, but this was soon nullified when Carlisle responded with another successful kick for the Warriors.
The second half was more evenly-matched but a try still looked unlikely. And as the game entered its final 10 minutes it was evident it would take something special - or a mistake - to settle things as both sides still preferred the kicking option.
With six minutes remaining, Wasps were penalised at a ruck 45 metres out and Carlisle made no mistake with a superb kick under pressure to put Worcester 12-9 ahead.
But two minutes from time, the visitors produced a try out of nowhere.
They won a line-out on the home 22 from which Simpson produced a searing break to take his side under the Warriors' posts. And from the resulting maul Birkett crashed over for the crucial points.
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