The suspects stripped the roof of St Laurence's in Foxton, Cambridgeshire, in the early hours of Tuesday.
However, they fled empty-handed after the van became bogged-down under the lead's weight.
Reverend Angela Melaniphy said heavy rain on Monday which drenched the graveyard turned out to be a "Godsend".
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Cambridgeshire Police is appealing for witnesses.
Although the roof was badly damaged by the thieves, the church was "fortunate" still to have its lead, the Reverend said.
"I've been laughing quite a lot, and, as one of our parishioners said, we thought the rain on Monday was a terrible nuisance, but it turned out to be a Godsend."
However, she added: "It is as if someone has violated where we worship... this house was built in honour of God, so that makes you sad."
It is thought the offenders drove up a gravel path at the rear of the church and stripped about £20,000 worth of lead from part of the nave, before abandoning the vehicle.
Eight churches within the Ely Diocese have been targeted by lead thieves so far this year, buildings officer Geoffrey Hunter said.
"It would be less of a financial hardship for the church simply to write cheques to the criminals for the price of the lead, as it costs us ten times as much to rectify the damage," he added.
Thieves trying to steal £20,000 of lead from a church roof were thwarted after the weight of the metal caused their getaway van to become stuck in the mud.