"I felt much more comfortable on court today," said Burdekin, who had to come from a set down to win his first round match on Tuesday. "It's never easy coming back from injury, but I'm pleased with today and will have to step it up to play Lucas. I've beaten him twice this season, but he's one of the top players out there."
Sithole beat Brit Antony Cotterill 6-2, 6-1, while world No. 1 David Wagner also came through against British opposition, defeating Richard Green 6-1, 6-2. Wagner faces Canadian third seed Sarah Hunter in the semi-finals after Hunter powered past Israel's Yosi Genish 6-0, 6-0.
All eight seeds are safely through to both the men's and women's singles quarter-finals after four of the eight men's second round matches ended in 6-0, 6-0 victories. Among those to progress to the last eight without dropping a game was British No. 1 Gordon Reid. World No. 4 Reid beat Israel's Asaaf Stokol and will now take on French eighth seed Frederic Cattaneo, who beat Brit Alex Jewitt 6-3, 6-3.
World No. 2 Stephane Houdet, world No. 3 Maikel Scheffers and world No. 8 Michael Jeremiasz also posted 6-0, 6-0 wins. Scheffers, the 2010 and 2011 British Open champion will play Frenchman Jeremiasz in the last eight, while Houdet will take on Dutch sixth seed Ronald Vink after his 6-3, 6-2 win over Brit Marc McCarroll.
The other men's quarter-final will see fourth seed Gustavo Fernandez of Argentina play fifth seed Joachim Gerard of Belgium after Fernandez beat Britain's world No. 1 junior Alfie Hewett 6-0, 6-1 and Gerard ended the challenge of Nottinghamshire's British No. 3 David Phillipson 6-2, 6-1.
Meanwhile, Britain's top two players Lucy Shuker and Jordanne Whiley secured straight sets wins to reach the women's singles quarter-finals. Sixth seed Shuker beat Colombia's Angelica Bernal 6-0, 6-1 and Whiley defeated Christine Schoenn of France 6-3, 6-3, as the duo advanced to women's singles quarter-finals against world No. 1 Sabine Ellerbrock and world No. 2 Aniek van Koot respectively.
Ellerbrock opened her account with a 6-2, 6-2 win over Japan's Miho Nijo, while 2012 British Open runner-up van Koot defeated Brit Lauren Jones 6-1, 6-0.
Dutch third seed Jiske Griffioen and fifth seed Kgothatso Montjane will go head-to-head for a place in the semi-finals after both easing though their opening matches against Israel's Veronica Zvjk and Turkey's Busra Un, respectively, without dropping a game.
"Today wasn't the toughest match, but the next one will be much tougher and all I can do is go out there and do the best I can like the previous times I've played here and hopefully I can get the result," said South Africa's Montjane.
The third women's singles match of the day to end 6-0, 6-0 was won by Chilean eighth seed Francisca Mardones, who beat 12-year-old Brit Esperanza Merry, the youngest player in the women's draw. Mardones will play fourth seed Marjolein Buis of the Netherlands for a place in the semi-finals after Buis defeated British No. 3 Louise Hunt 6-2, 6-1.
Wednesday's second day of play also saw the start of the men's, women's and quad doubles, with Burdekin and Sithole joining all-British duo Green and Ian Payne in the quad doubles semi-finals and fourth seeds McCarroll and Reid joining Hewett and Phillipson among three all-British pairings to reach the men's doubles quarter-finals.
Organised by the Tennis Foundation, the British Open Wheelchair Tennis Championships is one of six events to have Super Series status, the highest tier of tournament on the NEC Wheelchair Tennis Tour outside of the Grand Slams.