The Bears were making their World Cup debut and a good start saw them take a 3-0 lead through a Yuri Kushnarev penalty.
But the USA enjoyed the better of the contest, with Saracens full-back Chris Wyles kicking two penalties as well as converting Mike Petri's try.
However, Eddie O'Sullivan's men were given a late scare as a Konstantin Rachkov penalty narrowed the gap, before Russia rumbled into their 22 in the closing moments.
A converted try would have levelled the score, but the USA held out.
Clever, whose side had a short turnaround after losing to Ireland on Sunday, told ITV: "I was very nervous. They are a tough team and they are improving, every time we play them they are getting better and better. I am just so proud of the guys that we were able to finish that.
"It's an awesome feeling (to win). I am proud of the boys for sticking through. We made it tough on ourselves but the crowd got us through. Mentally we were there but physically we were a little slow off the cuff, but we have a break before our next game so hopefully we can be back to 100%."
The Eagles next face the daunting challenge of Australia in eight days' time.
Clever added that he is looking forward to facing a side of the quality of the Wallabies, now that his side have a win under their belt.
He said: "That will be a tough game and we are looking forward to a tough challenge against the big names. We want to improve every game and we look forward to that."
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