Calleri, who was playing for Argentina at the Olympics two weeks ago, had not done much to impress, but was almost clean through soon afterwards.
Only a fine tackle from Geraldo Alves denied him a sight of goal.
Lung then made a full-length save from Calleri, and then got some help from Michail Antonio, who got in the way of a shot from Byram to deflect it over the bar.
West Ham had finally built up a head of steam but it wasn’t enough as Astra held on for a famous win.
❑ WEST HAM have signed Swiss midfielder Edimilson Fernandes from FC Sion for £5m.
WEST HAM (4-3-3): Randolph; Byram, Reid, Ogbonna, Burke (Collins 88); Nordtveit (Valencia 46), Kouyate, Obiang; Tore, Antonio, Calleri (Fletcher 61).
Subs: Adrian, Oxford, Browne, Diangana.
ASTRA (4-2-3-1): Lung; Sapunaru, Alves, Fabricio, Morais; Seto, Lovin (Florea 88); Ionita (Stan 75), Niculae (Oros 57), Teixeira; Alibec.
Subs: Gavrilas, Raileanu, Moise, Balaure.
Referee: Manuel Grafe (Germany).