Reading moved seven points clear of the Women's Super League's bottom spot as Amy Turner's own goal gave them a victory over Spurs.
Turner nodded in during a mix-up with Spencer after the keeper tipped a long-range shot into the net.
Despite being beset by a flu outbreak, Reading dominated the game in the first half and would have extended their lead but for Spencer's save.
Spurs looked sharper after the break, but were indebted to Spencer again when she denied Sanne Troelsgaard from close range.
Reading was tired in the last part of the game and had keeper Jacqueline Burns to thank for keeping the game close.
Kelly Chambers' side climbed one place to ninth with their first clean sheet and second win of the season.
Harries, a 20-year-old youth product captaining Reading for the first time, was a persistent threat and had begun to celebrate when Spencer pulled off one of several early stops.
Three defeats in four WSL games puts Spurs at risk of being cut adrift from the top six.
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