Bicknell, 35, reached the landmark when he had Matthew Dennington edging one of his typical outswingers through to wicketkeeper and Surrey captain Jon Batty.
Afterwards Bicknell, whose four Tests for England spanned 10 years between 1993 and 2003, said: "I had the record in my sights at the start of the season and thought it was likely that I'd be in with a chance here at The Oval, so to do it here is fantastic.
"I'm absolutely delighted to have achieved this feat and I'm looking forward to playing for a few more years yet."
Surrey cricket manager Steve Rixon, the former Australia wicketkeeper and New South Wales and New Zealand coach, said: "Martin works tremendously hard at his game and his fitness, to take 1000 first-class wickets is an amazing achievement. He has bowled some great spells this summer, reminding us of why he was picked for England at the end of last season.
"Bicknell is a great role model for any young bowler, both in terms of his approach to training and bowling action.
"He is a modern day Surrey legend."
Bicknell is only the third current player to have taken 1,000 first-class wickets alongside Pakistan leg-spinner Mushtaq Ahmed, playing in England for County champions Sussex and Phil DeFreitas, the former England seamer now in his second spell with Leicestershire.