The commission said the security threat to the electoral process had been contained.
"The suspension of the counting and due announcement of results has been lifted with immediate effect," commission chair Mwangala Zaloumis told a press conference.
Zaloumis did not say if the theft of the ballot papers would affect the final results.
President Hakainde Hichilema is seeking a second term in the southern African country, a major copper producer, and is widely thought to have the edge over opposition leader and first-time presidential candidate Brian Mundubile.
The results are due to be announced on Monday.
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.
Zaloumis said the first set of results would be announced later on Friday and urged all candidates to "desist from declaring themselves as the winners of these elections," adding that it is an offence.
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.
