Andrus walked to begin the 14th, stole second and went to third on Jurickson Profar's sacrifice bunt, but Berkman and Adrian Beltre both grounded out.
"We kept putting ourselves in position to get something done and it didn't happen," Washington said. "But I'll keep taking us putting ourselves in position. Pretty soon, we'll break through."
Darvish allowed two earned runs and five hits. He walked three and struck out seven.
Blue Jays left-hander Mark Buehrle allowed one run and four hits in seven innings.
Jeff Baker homered off Buehrle in the seventh, the first homer off Buehrle since May 6.