The connected dominating set (CDS) concept has recently emerged as a promising approach to the area coverage in wireless sensor network (WSN). However, the major problem affecting the performance of the existing CDS-based coverage protocols is that they aim at maximizing the number of sleep nodes to save more energy. This places a heavy load on the active sensors (dominators) for handling a large number of neighbors. The rapid exhaustion of the active sensors may disconnect the network topology and leave the area uncovered. Therefore, to make a good trade-off between the network connectivity, coverage, and lifetime, a proper number of sensors must be activated. This paper presents a degree-constrained minimum-weight extension of the CDS problem called DCDS to model the area coverage in WSNs. The proper choice of the degree-constraint of DCDS balances the network load on the active sensors and significantly improves the network coverage and lifetime. A learning automata-based heuristic named as LAEEC is proposed for finding a near optimal solution to the proxy equivalent DCDS problem in WSN. The computational complexity of the proposed algorithm to find a 11-∊ optimal solution of the area coverage problem is approximated. Several simulation experiments are conducted to show the superiority of the proposed area coverage protocol over the existing CDS-based methods in terms of the control message overhead, percentage of covered area, residual energy, number of active nodes (CDS size), and network lifetime.