Billy Jones missed the best chance of the match but the Black Cats are now seven points away from safety with only ten matches of the campaign remaining.
Rodwell sliced an early effort wide for the home team while at the other end George Boyd mis-kicked when well-placed.
Scott Arfield’s free-kick curled just wide of Jordan Pickford’s goal as the Clarets began to get on top midway through the first half while Sean Dyche’s team had penalty appeals rejected when Boyd appeared to have been brought down.
Darron Gibson lashed a powerful effort over the bar on the half-hour mark but Burnley should have taken the lead five minutes before the interval when Ashley Barnes side-footed his effort wide after being set up by the lively Arfield.
David Moyes’ side started the second half brightly but, after Bryan Oviedo had played him in, Fabio Borini’s weak shot was easily gathered by Tom Heaton.
Adnan Januzaj then fired straight at the Clarets goalkeeper before Seb Larsson lashed a shot over the bar when he should really have hit the target.
Jones somehow contrived to head wide on the hour mark from four yards out as the Black Cats searched for a crucial goal.
Januzaj’s goalbound effort was blocked by Ben Mee but it was Burnley that went closest in the final stages as they tried to clinch their first away win of the season, with Sam Vokes nodding Robbie Brady's free-kick wide before the Wales international was denied by a superb Pickford save.