FIN and RST packets that close TCP connections are often delayed by timeout. In cellular networks, delayed FIN/RST packets often incur significant energy consumption overhead for handsets. On the other hand, closing TCP connection immediately after its last data transfer avoids the energy overhead, but can cause performance degradation as doing so makes reusing TCP connections difficult. To resolve such a dilemma, we propose a novel TCP extension called STC (Silent TCP connection Closure) using which both endpoints close a TCP connection silently without exchanging FIN or RST packets after timeout. Our solution is lightweight, backward-compatible, and incrementally deployable. It requires modifications to smartphone operation systems, but, if supported by cellular middleboxes, no change to remote servers. We evaluate the benefits of STC using a 10-day real trace consisting of 0.6 million LTE user sessions. When fully deployed, STC can save the overall handset radio energy consumption by up to 11.3% and reduce the network-wide signaling load by up to 6.0%.