The 29-year-old ended a run of 14 Premier League games without a goal to give Everton an early lead.
And it is understood to be Walcott's first strike since his beloved grandad passed away earlier this year.
The English star slotted the ball past Hugo Lloris from close range after an assist from Dominic Calvert-Lewin.
He looked close to tears as he dropped to his knees and pointed at the sky to celebrate.
In November, the ex-Arsenal star wrote a heartfelt message alongside a picture of himself and his grandad on Instagram.
It read: "I can't thank you enough for being the heart and soul of our family Grandad.
"You can now rest in peace watching over all of us. You will be truly missed."
But Walcott's goal was cancelled out just six minutes later when Son Heung-min bagged the equaliser from a tight angle.
Dele Alli then put Mauricio Pochettino's men ahead before Harry Kane and Christian Eriksen scored to make 4-2.
Gylfi Sigurdsson pulled one back to give Everton hope, but Son and Kane hit back again to make it 6-2.
Walcott penned a three and a half year deal with the Toffees in January after 12 years with Arsenal.
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