The fifth-seeded Australian Open champion last reached the fourth round at the English competition in 2009.
Wawrinka, 28, served a dozen aces and won 83 percent of his first service points en route to the third round victory over Istomin.
The match, originally set for Saturday, was rescheduled due to rain.
The Lausanne native will now face Spain's Feliciano Lopez for a place in the quarterfinals.
Wawrinka's victory came two days after fellow countryman Roger Federer decisively beat Colombian Santiago Giraldo in three straight sets.
Federer faces a fourth round showdown on Tuesday against Tommy Robredo, in a rematch of their 2013 US Open encounter, which was won by the Spaniard in straight sets.