She was the only Scot to survive the first round at Royal St George's GC and is one of only five British players to reach the last 16.
Also in the that round are Alice Hewson (Berkhamsted), Londoner Rosie Davies (Indiana University), Hayley Davis (Ferndown) and Gabriela Cowley (Hanbury Manor). Dryburgh and Cowley are the only survivors from the GB & Ireland team which competed in the Curtis Cup earlier in the month.
Dryburgh had established a two-hole lead over Beck, also a student at an American university, by the 14th tee. Beck was able to pull one hole back but the Scot kept her nerve to secure victory at the 18th. She will next play Denmark's Emily Pedersen, the holder of the European women's amateur title and player ranked No.26 in the world.
Lauren Whyte (St Regulus), meanwhile, lost by two holes in the first round to Davies, while Alyson McKechin, last year's Scottish champion from Elderslie, lost 2 and 1 in the first round against Hayley Davis.
Connie Jaffrey, who lost in the final of the Scottish at Prestwick last month, was beaten 3 and 2 by Spain's Luna Sobron, who reached the final of this championship 12 months ago.
Ailsa Summers, of Carnoustie Ladies, also fell at the first hurdle after being beaten by one hole against Ann Van Dam (Netherlands), who advanced to the last 16 by seeing off Sophie Lamb (Clitheroie) 4 and 3. Annabel Dimmock, the Curtis Cup player from Wentworth, lost unexpectedly against Italy's Bianca Maria Fabrizio in the first round.