Divock Origi pounced to sweep in from close range and level the scoreline before half-time after Nolito had clipped the Spanish side into the lead.
Sevilla were reduced to 10 men with 15 minutes remaining when Joris Gnagnon was shown a straight red card for a wild challenge on Yasser Larouci.
However, they snatched victory in the final minute when Alejandro Pozo rounded Simon Mignolet and slotted into the empty net.
The teamsheet confirmed Jordan Henderson, Virgil van Dijk, Georginio Wijnaldum, Trent Alexander-Arnold and Andy Robertson were to be handed their first starts of the summer and Andy Lonergan was given a debut in goal as Fenway sweltered in temperatures well in excess of 30 degrees.