Final official results from the election, which coincided with an approaching typhoon, are expected early on Monday.
The US-drafted constitution's Article 9, if taken literally, bans the maintenance of armed forces. But Japanese governments have interpreted it to allow a military exclusively for self-defence.
Backers of Abe's proposal to clarify the military's ambiguous status say it would codify the status quo. Critics fear it would allow an expanded role overseas for the military.
Abe said he would not stick to a target he had floated of making the changes by 2020. "First, I want to deepen debate and have as many people as possible agree," he told a TV broadcaster. "We should put priority on that." The LDP's junior partner, the Komeito, is cautious about changing the constitution, drawn up after Japan's loss in World War Two. Several opposition parties favour changes, but don't necessarily agree on details.