Sanchez has joined, subject to international clearance, for an undisclosed fee on a four-year deal.
The 28-year-old, nicknamed "The Rock", played for his country at this summer's World Cup in Brazil and has 44 caps for the South American entertainers.
While the fee was not noted, it is understood to be in the region of £4.7m.
"We have to improve on what we did last season. That goes without saying," Lambert told reporters. "We have a stronger squad this season, you can sense it.
"When we beat Chelsea [in March] we were sitting 12th. Then, all of a sudden, we couldn't get a win.
"We lost Libor Kozak and Christian Benteke too. That certainly hurt us. The run-in was nowhere near good enough. We have to do a lot better."
Sanchez only spent one season at Elche, who he helped keep in La Liga following their promotion.
Previously he turned out more than 200 times for French side Valenciennes, having come through the youth ranks of Uruguayan teams Danubio and River Plate.
Sanchez is a strong, combative anchorman, but he can also play a bit. He almost moved to England two years ago, and will give Villa a much-needed streetwise element to their midfield, which was naive at times last season. £4.7m is a bargain although, while a class act for Colombia, he has only turned out for small clubs in the past. How will he handle the pressure?
- Sports & Recreation
- Paul Lambert