Their medal followed Samantha Murray's individual gold and the women's team silver earlier in the week.
After strong fencing, swimming and show jumping rounds, French and Evans took an 18-second advantage into the final shoot-run combined phase.
However, Lithuania's team overtook them and won gold by just three seconds.
There was no shame in defeat, though, with Laura Asadauskaite being the reigning Olympic champion and Justinas Kinderis having won the men's world title last year.
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"With Laura behind me I was running for my life, I was really scared," said 23-year-old French, who won silver with Murray in the women's relay.
"It's been a really good day though."
The British pair were particularly impressive in the morning's fencing round, winning 27 of their 38 battles to rank second.
They climbed to the top of the rankings with a strong 2x100m freestyle relay swim before both delivered clean show jumping routines.
Although French maintained Britain's advantage heading into the changeover in the combined event, 21-year-old Evans struggled in the second shooting phase which allowed Kinderis to move clear.
"Me and Justin were playing mind games the whole way around," Evans told BBC Sport.
"Unfortunately I didn't managed to pull it off today, but I'm still happy to have won a silver medal."
The mixed-team event is not currently part of the Olympic programme, but the sport's international governing body - UIPM - hope it could be included for the 2020 Games in Tokyo.
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The introduction of the Fixed Term Parliaments Act, however, means that even if she wants to, Mrs May might not find it so easy.
The Act was introduced after the 2010 election to bind together the two parts of the new coalition government.
The discretion of prime ministers to pick the timing of elections was removed and the dates fixed five years apart - we are next due one on Thursday 7 May 2020.
The Act provides two mechanisms by which an early election could be triggered, which are:
Taking each in turn, the government does not have the numbers on its own to achieve a two-thirds majority so any vote would depend on what the other parties do.
Jeremy Corbyn has warned the Labour Party to prepare for an election next year and Newsnight reported in September that he planned to instruct his MPs to vote in favour of an early election if the government wanted to call one (although that has not been confirmed). Latest polls suggest that supporting an early election would be a mistake for Labour.
It is unlikely the Scottish National Party would come to the government's assistance in bringing about an early election.
Looking at the second mechanism, if other parties do not support an early election it would be unprecedented in British politics for a government to bring about its own downfall (though such engineering is not unknown elsewhere) and the government voting no confidence in itself could be a risky strategy.
More likely is that the Fixed Term Parliaments Act could be repealed or amended, to require, for example, a simple majority of MPs in favour of an early election, though the House of Lords may have something to say about this.
An amendment to existing legislation seems more straightforward than trying to reintroduce the prerogative power to call elections.
In short, the calling of an early election would require either a change in the law, or some unusual parliamentary gymnastics.
Kate French and Joe Evans won Britain's third medal at the Modern Pentathlon World Championships in Poland by winning silver in the mixed team relay.
[NEXT_CONCEPT]
With the permission of the Queen, prime ministers used to be able to call a general election at a time of their choosing.