The 80s are generally considered a time when many rock bands who originated in the 1960s or 1970s lost the plot and put out subpar material.
What are some 80s records by 60s/70s bands that actually live up to the band’s past work?
As for the Stones … none of the three albums worth mentioning at all were close to their “best work”: I find Tattoo You (1981) overrated; Steel Wheels (1989) was a happy surprise, but no great shakes. That leaves Emotional Rescue (1980), which I really like – but it barely fits your criteria.
I think the Kinks’ Word of Mouth (1984) comes closer to what the OP is after.
90125 immediately comes to mind, as pretty much a re-invention of Yes.
Jethro Tull’s Crest of a Knave is well-known as being better than Metallica. 
Aerosmith had Permanent Vacation in 1987 and* Pump* in 1989. Two albums which I think are the best work.
I feel that Give the People What They Want (1981) was one of their best albums.
Some other pre-eighties artists who had top-notch albums in the eighties:
David Bowie - Let’s Dance (1983)
Z.Z. Top - Eliminator (1983)
Bruce Springsteen - Born in the U.S.A. (1984)
John Fogerty - Centerfield (1985)
Paul Simon - Graceland (1986)
If we’re doing pure 70s (rather than 60s-70s), then Michael Jackson is an obvious example.
Excellent pick – he’s totally a 60s artist as well as a 70s one.
*Momentary Lapse of Reason *by Pink Floyd
*Graceland *is a great choice.
Cloud 9 by George Harrison
Traveling Wilburys
90125 by Yes
The Other Side of Life by the Moody Blues is pretty well regarded even though it is more synth pop than the MBs usual Progressive Rock.
I prefer Genesis post Gabriel but Pre Invisible Touch
Me too. Duke, Abacab, and 3 Sides Live are great.
Also State of Confusion. “Come Dancing” was a big hit, and deserved to be (it’s what first made me notice The Kinks).
I actually prefer their previous (80s) albums, Long Distance Voyager and The Present.
Paul McCartney’s Tug of War, at least, is a good album.
The Beach Boys’ self-titled 1985 album isn’t objectively their best work, and production-wise it sounds very 80s, but it has a special place in my heart because to me, it was their first new release, and I saw them live that year.
Stevie Wonder was past his prime but still putting out worthwhile records.
Would you count Roxy Music and Avalon. It was their final album, released in 1982, and I’d say one of their best works.
Roy Erickson got known in the 1960s because of the 13th Floor Elevators, but his 1986 solo album Don’t Slander Me totally fucking rocks!
I still think Deep Purple’s 1984 album Perfect Strangers was a decent album.
To me The Clash is IMO indubitably a '70s band, but 1982’s Combat Rock remains the 2nd greatest rock album yet recorded (also IMO).
9 years after Radar Love was all over American radio stations, Golden Earring led us into their Twilight Zone.
Van Halen’s 1984 was their biggest album yet and was simply inescapable on radio (and EmptyVision) for over a year.
Supertramp struggled after the 1979 smash hit Breakfast In America, enduring not just creative lulls but also changes in membership. Still, I have a fondness for 1985’s Brother Where You Bound; here’s Cannonball.
Manfred Mann first topped the charts in 1964 with a cover of The Exciter’s Do Wah Diddy Diddy. In 1977, he topped the charts again, this time with a cover of Bruce Springsteen’s Blinded By The Light by Manfred Mann’s Earth Band. In 1983, he was all over the airwaves again, thanks to a cover of The Police’s Demolition Man.
And of course George Clinton arguably peaked in 1982 when he gave us Computer Games. The world is a better place with Atomic Dog, yo.
King Crimson put out three good albums in the 80s. Not as good as their previous output, but still good.
Some others that might qualify:
Dire Straits (formed in late 70s)
Journey
REO Speedwagon
Black Sabbath
Ozzy Osborne (separately)
Heart (some hits in the 80s but their best was clearly in the 70s)
Cher
It’s a matter of personal opinion. Mine is that “Come Dancing” is a great song. But I feel as an overall album, Give the People What They Want is better than State of Confusion.
Acoustic guitarist Leo Kottke’s most famous work was in the early '70s, but his '80s album Time Step is considered by some to be on that same level.
Here’s one more from the folk world: Ronnie Gilbert was one of the Weavers in the 50s and early 60s, but she enjoyed a resurgence (creative output, not just reunion-tour-type stuff) in the mid-80s, including as a duet with the younger Holly Near.
Fellow travelers 
Odetta and Pete Seeger were also putting out new, varied, vital music in the 1980s.
Lou Reed’s “New York“ from 1989 is widely considered his strongest solo album, including by me.