Sunderland paid the price for failing to convert a number of chances in a one-sided first half as Sordell struck twice in nine minutes to book Bolton a home clash with Everton.
Wanderers were on the back foot early on as Stephane Sessegnon had an effort deflected wide before Craig Gardner cleared the bar with a rasping drive.
Adam Johnson drilled wide and then had another well-struck shot tipped over by Andy Lonergan, while Simon Mignolet easily dealt with a low effort from Sordell.
Lonergan did well to palm away a superb free-kick from Johnson, and Connor Wickham nodded straight at the keeper as the hosts ended the half without reward.
Bolton looked a different side after the interval as Mignolet got down well to deny Sordell before Chris Eagles blazed over.
But the visitors broke the deadlock on 64 minutes when Darren Pratley was scythed down by Jack Colback in the box and Sordell smashed home the resulting penalty.
The Sunderland defence contributed to their own downfall nine minutes later as they failed to deal with a Tyrone Mears cross and Sordell turned sharply to slot the second past Mignolet.
The home side responded as Phil Bardsley drove wide and Matthew Kilgallon headed over from close range, but Bolton held out to claim a notable Premier League scalp.