David McGoldrick had the best of the first-half chances at Bramall Lane, firing Lundstram's cross straight at Vicente Guaita from close range shortly before half-time.
The home side then took the lead two minutes after the break, Guaita parrying a Luke Freeman effort for Lundstram to find the net.
Andros Townsend forced Dean Henderson into a low save six minutes later as Palace searched for an equaliser, but the hosts held on as they continued their unbeaten start to their first Premier League campaign since 2006/07.
The win puts the Blades in seventh position with four points, while 14th-placed Crystal Palace have one point from two matches.
Sheff Utd: 24 Aug v Leicester (H)
Crystal Palace: 24 Aug v Man Utd (A)