Smith, the European champion, beat Gabriel Clemens 5-4 in Group O but needed a wider margin after losing 5-3 to Rusty-Jake Rodriguez on Saturday.
He missed two match darts at double 18 in the eighth leg before winning the decider, and Clemens must now beat Rodriguez 5-3 or better to go through, with Rodriguez needing four legs.
Greaves had to beat Niels Zonneveld 5-2 to stay in contention in Group A and fell 3-1 behind before winning four legs in a row for a 5-3 victory.
Zonneveld needs to beat Luke Littler 5-2 or better on Monday to deny the defending champion a place in the last 16.
Martin Schindler whitewashed Jan Schmidt 5-0 in Group N, taking the fourth leg even though Schmidt had opened it with six perfect darts, and plays Chris Dobey on Tuesday for a place in the knockout stage.
Paolo Nebrida beat Joe Cullen 5-2 in Group I, leaving Cullen needing a win over Luke Humphries in his final match.
Sebastian Bialecki knocked out Joe Hunt with a 5-4 win in Group E, taking out 78 in the deciding leg, and meets Gerwyn Price next. Hunt hit 12 140s in defeat.
Bradley Brooks hit four 180s in a 5-3 win over Carl Sneyd that eliminated the debutant from Group M, and faces Gian van Veen on Tuesday.
Ricky Evans beat Cam Crabtree 5-4 in Group K, winning the last two legs after Crabtree had taken out 146 on his way to a 3-1 lead. Crabtree is out, and Evans plays James Wade in his final match.
Henry Coates won the last two legs to beat Jeffrey de Graaf 5-4 in Group L, a day after missing two match darts against Michael van Gerwen.
Littler's meeting with Zonneveld is one of several group deciders on Monday. Rob Cross plays Anton Ostlund for top spot in Group G after averaging 114 in his win over Jonny Clayton, while Kevin Doets and Tommy Morris meet to decide Group C following Josh Rock's exit.
Stephen Bunting faces Cameron Menzies, Ryan Searle meets Andrew Gilding, Wessel Nijman takes on Niko Springer and Jermaine Wattimena plays Alan Soutar, with the winner of each match going through to the last 16.