Sanchez this week was forced to leave a Chile training session early with a muscular issue and Rueda criticised Inter's management of the attacker's physical condition.
The club has not hidden its feeling that its players should not be travelling for international matches during the global coronavirus pandemic, claiming the games often lead to injuries amid a congested schedule.
Rueda accused the club of a lack of respect and the Nerazzurri responded by calling his comments "unacceptable, offensive and not corresponding to reality".
Sanchez started on the bench for Saturday's (AEDT) 2-0 victory over Peru in the CONMEBOL 2022 FIFA World Cup qualifying section and was sent on in the 84th minute, replacing Felipe Mora.
"We talked with Alexis and the medical team about his situation. If we could give him 15 or 20 minutes it was very important," Rueda said.
"He came on and played well in a key stage of the game. It was important for him to gain continuity and confidence without taking risks."
Chile's win was secured by a pair of first-half goals from captain Arturo Vidal, the first of which was a sensational 25-yard strike that flew into the top-right corner.
⚽️ GOAL!! OH YES 🔥🔥🔥 A truly magnificent opener from @kingarturo23 😲@laRoja 1️⃣-0️⃣ @TuFPF— beIN SPORTS (@beINSPORTS_AUS) November 13, 2020
Watch LIVE on CONNECT @kayosports, @Foxtel & Fetch!! ▶️ https://t.co/eUHe56iYjS#CONMEBOL | #WCQ | #CHIPER | @bet365 pic.twitter.com/k5eyRmLaTE
"He assumed the captaincy with great integrity. The dressing room respects him, because of his experience, the heart he has," Rueda said.
"Arturo's decision and daring to risk allowed us to embrace one another, but Peru never made the game or the possibility to score easy for us."
Vidal's double moved him onto 31 international goals, taking him clear of Carlos Caszely for sole ownership of fifth in La Roja's all-time scoring chart.
However, with Chile having taken one point from their opening two qualifiers it was the win that pleased him more.
"I'm happy for that [surpassing Caszely], but the most important thing is the team, fighting to go to another World Cup. That is more important than the goals," Vidal said.
"I want to congratulate my team-mates for the game we played, against a very strong team which have been playing very well in recent years.
"The first half we played much better, we looked to control the game. The second cost us a bit, we backed down a little, but the important thing was to get the three points and gain confidence."
Peru was unable to recover and record a third straight win against Chile for the first time, having triumphed in the previous two encounters 3-0.
It has now lost six of its past seven against La Roja in FIFA World Cup qualifying games.