After Darion Radcliff's RBI single tied the score, Houston drove a line drive down the left field line to bring in pinch-runner Lawrence Noble from second base.
Cumberland scored five runs in the bottom of the third to take a 7-6 lead, but couldn't score in the last two innings and was eliminated.
Chicago will play Pearland, Texas, on Tuesday night.
Texas 11, Washington 4Layne Roblyer hit his second home run of the tournament, helping Pearland, Texas, to an 11-4 victory over Lynnwood, Washington.
Roblyer, who homered against Philadelphia on Sunday, had a three-run shot in the third inning against Lynnwood, which was eliminated.
Pearland (2-1) scored early, getting a run in the first inning followed by Roblyer's shot to the lawn seats in right field. Christian Terranova scored on a passed ball in the fourth inning.
Pearland led 5-0 before Lynnwood (1-2) got on the scoreboard with three runs in the bottom of the fifth.
Two doubles and a single from the Northwest regional champions made the deficit reasonable, but Pearland scored five runs on two wild pitches, a passed ball and a single in the sixth to seal the win.
Pearland will play the winner of Chicago and Cumberland, Rhode Island, on Tuesday night.
South Dakota 5, Czech Republic 3Daniel Vigoren hit a go-ahead double as Rapid City, South Dakota took advantage of two errors to beat the Czech Republic, 5-3, at the Little League World Series on Monday.
Colton Hartford had scored the tying run when he got in a rundown between third and home and Czech Republic pitcher Tomas Oppelt made a throwing error to third base.
Vigoren, who followed with a line-drive double to the right-field gap, later scored on an overthrow by Czech Republic shortstop Marek Krejcirik, who had homered in the game's first at-bat.
Czech Republic's Miroslav Krivanek tied the game at 2 in the top of the fifth with a bloop RBI double into shallow right field. Krivanek took third on the throw and scored on a wild pitch moments later.
Rapid City took the lead in the third inning with a two-run double into the left-field gap by Matthew Hegre.
Mexico 6, Australia 2Erick Vela hit a three-run home run in the fourth inning as Mexico eliminated Australia, 6-2.
Vela stepped to the plate with one out and men on second and third. He took the first pitch he saw deep to right field to put Mexico ahead.
Aldair Tellez pitched 5 2/3 innings for Mexico, striking out 12.
Mexico will play the winner of Puerto Rico and Venezuela on Tuesday afternoon.
Venezuela 2, Puerto Rico 1Ronny Medina struck out nine in five sparkling innings, and Venezuela beat Puerto Rico 2-1 on Monday in an elimination game in the Little League World Series.
Yadiel Santana scored on a passed ball in the first, but that was it for Puerto Rico. Medina allowed two hits.
Venezuela next plays Mexico on Tuesday afternoon.
Venezuela scored in each of the first two innings. Julio Rejon homered in the first, and Jorge Cabrera swiped third in the second and went on to take home when catcher Alvin Martinez overthrew third on the play.
The Associated Press contributed to this report.
Illinois Eliminates Rhode Island
Josh Houston drove in the go-ahead run for Illinois as they defeated Rhode Island 8-7, eliminating them from the Little League World Series.
Kids Receive Speech To Remember
Rhode Island Little League coach Dave Belisle delivers a meaningful speech to his players after their loss to Illinois.
Mexico Dispatches Australia
Mexico rallied to beat Australia 6-2 and advance at the Little League World Series.
Venezuela Edges Puerto Rico
Puerto Rico loses to Venezuela, eliminated from Little League World Series.