President Hakainde Hichilema is seeking a second term in the southern African copper producer and is widely thought to have the edge over opposition leader and first-time presidential candidate Brian Mundubile.
It was unclear whether the suspension would delay the announcement of results, which were previously due on Monday.
"The violence is targeted at poll staff, and in some instances has resulted in the theft of marked ballot papers in ballot boxes," chief electoral officer Brown Kasaro told a briefing on Friday in the capital Lusaka, adding that the commission would review the suspension within 24 hours.
Earlier, Hichilema had called for patience in a post on social media, while Mundubile claimed he had won, without providing evidence, and called for the results to be released immediately.
The electoral commission has not yet announced any official results, although some preliminary counts have been released at constituency level.
"Based on information transmitted to us from polling stations and ... centres across the country, our tabulation indicates that we have won this election," Mundubile said in a video posted on his social media platforms.
"We therefore view the continued delay in the transparent announcement and verification of presidential results with grave concern."
Hichilema wrote on social media shortly before the suspension of vote-counting was announced that he had received "encouraging results".
"Let us remain peaceful, patient and patriotic as we await the official outcome from those entrusted with managing the vote," he said.
Zambia holds regular multi-party elections which are usually peaceful, although there have been isolated instances of violence in previous polls.
Civil society groups and opposition parties had warned of growing repression in the lead-up to the election including restrictions on opposition campaigns, although the government denied this.
