Nuno Espirito Santo's side were taken to extra time before Summerville ended the underdogs' stubborn resistance with his sixth goal in his last seven games in all competitions.
West Ham's Freddie Potts was sent off soon after Summerville's strike in the first period of extra time, but the Premier League strugglers held on to survive a gruelling fourth-round clash.
With their team three points from safety in the Premier League, the Hammers' erratic display will hardly ease fans' fears about their chances of avoiding relegation.
West Ham supporters flew a plane over the Pirelli Stadium with a banner calling for David Sullivan and Karen Brady to sell the club, while supporters inside the stadium chanted "sack the board."
The East Londoners had been in better form of late, winning three of their last five league games and holding Manchester United to a draw on Tuesday.
Nuno made 10 changes as he prioritised West Ham's fight for top-flight survival.
With West Ham and Burton sitting in the relegation zones in their respective divisions, 47 spots separated the two sides in the English football pyramid.
But West Ham's start was so slow that they didn't manage a touch in the Burton penalty area for the first 20 minutes.
They were under duress through Tyrese Shade's low strike, which forced a good stop from West Ham keeper Alphonse Areola.
Thirty-seven minutes were on the clock by the time West Ham finally had a shot - a Callum Wilson curler that was held by Bradley Collins.
West Ham were marginally better after the interval.
Collins saved a towering Konstantinos Mavropanos header before Wilson's blast was tipped over by the Burton keeper.
Soungoutou Magassa was denied by Collins, but West Ham made the breakthrough five minutes into extra time.
It was a moment of magic in a contest largely devoid of inspiration as Summerville made a dynamic break down the left flank before smashing a deflected strike into the top corner from the edge of the area.
West Ham substitute Potts was dismissed for a reckless lunge on Julian Larsson in the 101st minute.
Jack Armer and Kyran Lofthouse should have equalised, but both wasted chances to force a penalty shoot-out.