Both LeMond and Chiappucci were in the pack that finished more than five minutes behind Konyshev on a hot day in France near the Spanish border.
Today's stage had the last major climb of the tour, a 5,609-foot peak just outside the starting point of Lourdes. The leg ended in Pau after a circular route of 93 miles.
In becoming the first Soviet citizen to win a stage of the tour, Konyshev covered the distance in 4 hours, 8 minutes and 25 seconds.
He and Johan Bruyneel of Belgium had pulled away from a breakaway group not long before the approach to Pau as the race moved out of the Pyrenees.
Konyshev, second to LeMond in last year's world professional road race, held off the challenge from the Belgian. Steve Bauer of Canada was third, 11 seconds behind.
LeMond suffered a puncture on the last climb of the Tour, the Col de Marie Blanque, just after 1988 Tour winner Pedro Delgado of Spain launched an attack.
LeMond lost two minutes on the bunch, which included leader Chiappucci of Italy, while he waited for a wheel change.
But his "Z" team stayed behind to help and worked hard to get him back among the main group again.
Apart from punctures, his main problem seems to be saddle-soreness. He confessed before the stage that he was suffering from a three-inch raw wound.
But he said it will not stop his finishing the Tour, and there has been no sign that it is handicapping him unduly.
He is not the only member of the walking wounded brigade in the race. There were rumors before the stage that Delgado, third overall, would not start because of intestinal problems.
The Spaniard, who had been surprisingly lacking in aggression in the mountain stages before today, said Tuesday that he has been suffering for several days.
The race goes into two flat days, starting with Thursday's 18th stage of 126 miles to Bordeaux, before the crucial time trial Saturday when LeMond should take the leader's yellow jersey--if he does not do so before then.