Flame interaction with obstacles can affect significantly its behavior due to flame front wrinkling, changes in the flame front surface area, and momentum and heat losses. Experimental and theoretical studies in this area are primarily connected with flame acceleration and deflagration to detonation transition. This work is devoted to studying laminar flames propagating in narrow gaps between closely spaced parallel plates (Hele–Shaw cell) in the presence of internal obstacles separating the rectangular channel in two parts (closed and open to the atmosphere) connected by a small hole. The focus of the research is on the penetration of flames through the hole to the adjacent channel part. Experiments are performed for fuel-rich propane–air mixtures; combustion is initiated by spark ignition near the far end of the closed volume. Additionally, numerical simulations are carried out to demonstrate the details of flame behavior prior to and after penetration into the adjacent space. The results obtained may be applicable to various microcombustors; they are also relevant to fire and explosion safety where flame propagation through leakages may promote fast fire spread.