"Los Chapines" finished third in Group F with three points, but having been outscored 15-8 that left it behind Thailand, Japan Egypt and fellow CONCACAF side Panama for a berth in the second round.
Estuardo De Leon gave Guatemala the lead in the ninth minute from a corner, but the Solomon Islands equalized on George Stevenson's goal six minutes later.
The Pacific islanders went ahead in the 17th when Jeffery Bule poked a ball through the legs of Guatemalan keeper William Ramirez, who kept his side close by blocking Moffat Sikwaae's penalty just before halftime after Jose Gonzalez was ejected for denying the original scoring opportunity.
Alan Aguilar pulled Guatemala even less than two minutes after the start of the second half.
However, Micah Leaalafa put the Solomons ahead again in the 29th and the Oceania side sealed the victory when keeper Anthony Talo punted into an open net in the 36th after Carlos Merida had been inserted for Ramirez as a fifth attacker.
Walter Enriquez pulled Guatemala within 4-3 with 12 seconds remaining, but it was not enough to prevent the Solomon Islands from winning its first match in seven tries at the Futsal World Cup.