diff --git a/icu4c/source/i18n/measunit_extra.cpp b/icu4c/source/i18n/measunit_extra.cpp
index b997167b311..e43ee6597d8 100644
--- a/icu4c/source/i18n/measunit_extra.cpp
+++ b/icu4c/source/i18n/measunit_extra.cpp
@@ -321,6 +321,14 @@ private:
 
 class Parser {
 public:
+    /**
+     * Factory function for parsing the given identifier.
+     *
+     * @param source The identifier to parse. This function does not make a copy
+     * of source: the underlying string that source points at, must outlive the
+     * parser.
+     * @param status ICU error code.
+     */
     static Parser from(StringPiece source, UErrorCode& status) {
         if (U_FAILURE(status)) {
             return Parser();
@@ -340,6 +348,10 @@ public:
 
 private:
     int32_t fIndex = 0;
+
+    // Since we're not owning this memory, whatever is passed to the constructor
+    // should live longer than this Parser - and the parser shouldn't return any
+    // references to that string.
     StringPiece fSource;
     UCharsTrie fTrie;
 
@@ -399,7 +411,6 @@ private:
         // 1 = power token seen (will not accept another power token)
         // 2 = SI prefix token seen (will not accept a power or SI prefix token)
         int32_t state = 0;
-        int32_t previ = fIndex;
 
         // Maybe read a compound part
         if (fIndex != 0) {
@@ -432,7 +443,6 @@ private:
                     fAfterPer = false;
                     break;
             }
-            previ = fIndex;
         }
 
         // Read a unit
@@ -449,7 +459,6 @@ private:
                         return;
                     }
                     result.dimensionality *= token.getPower();
-                    previ = fIndex;
                     state = 1;
                     break;
 
@@ -459,7 +468,6 @@ private:
                         return;
                     }
                     result.siPrefix = token.getSIPrefix();
-                    previ = fIndex;
                     state = 2;
                     break;
 
@@ -469,7 +477,6 @@ private:
 
                 case Token::TYPE_SIMPLE_UNIT:
                     result.index = token.getSimpleUnitIndex();
-                    result.identifier = fSource.substr(previ, fIndex - previ);
                     return;
 
                 default:
@@ -576,7 +583,7 @@ void serializeSingle(const SingleUnitImpl& singleUnit, bool first, CharString& o
         return;
     }
 
-    output.append(singleUnit.identifier, status);
+    output.appendInvariantChars(gSimpleUnits[singleUnit.index], status);
 }
 
 /**
diff --git a/icu4c/source/i18n/measunit_impl.h b/icu4c/source/i18n/measunit_impl.h
index cf0ea63d2af..9657ff2c1d9 100644
--- a/icu4c/source/i18n/measunit_impl.h
+++ b/icu4c/source/i18n/measunit_impl.h
@@ -69,9 +69,6 @@ struct SingleUnitImpl : public UMemory {
     /** Simple unit index, unique for every simple unit. */
     int32_t index = 0;
 
-    /** Simple unit identifier; memory not owned by the SimpleUnit. */
-    StringPiece identifier;
-
     /** SI prefix. **/
     UMeasureSIPrefix siPrefix = UMEASURE_SI_PREFIX_ONE;
     
diff --git a/icu4c/source/test/intltest/measfmttest.cpp b/icu4c/source/test/intltest/measfmttest.cpp
index 9306a5660bc..79eb9b461a0 100644
--- a/icu4c/source/test/intltest/measfmttest.cpp
+++ b/icu4c/source/test/intltest/measfmttest.cpp
@@ -82,6 +82,7 @@ private:
     void TestInvalidIdentifiers();
     void TestCompoundUnitOperations();
     void TestIdentifiers();
+    void Test21060_AddressSanitizerProblem();
 
     void verifyFormat(
         const char *description,
@@ -204,6 +205,7 @@ void MeasureFormatTest::runIndexedTest(
     TESTCASE_AUTO(TestInvalidIdentifiers);
     TESTCASE_AUTO(TestCompoundUnitOperations);
     TESTCASE_AUTO(TestIdentifiers);
+    TESTCASE_AUTO(Test21060_AddressSanitizerProblem);
     TESTCASE_AUTO_END;
 }
 
@@ -3445,6 +3447,36 @@ void MeasureFormatTest::TestIdentifiers() {
     }
 }
 
+// ICU-21060
+void MeasureFormatTest::Test21060_AddressSanitizerProblem() {
+    UErrorCode status = U_ZERO_ERROR;
+    MeasureUnit first = MeasureUnit::forIdentifier("one", status);
+
+    // Experimentally, a compound unit like "kilogram-meter" failed. A single
+    // unit like "kilogram" or "meter" did not fail, did not trigger the
+    // problem.
+    MeasureUnit crux = MeasureUnit::forIdentifier("one-per-meter", status);
+
+    // Heap allocation of a new CharString for first.identifier happens here:
+    first = first.product(crux, status);
+
+    // Constructing second from first's identifier resulted in a failure later,
+    // as second held a reference to a substring of first's identifier:
+    MeasureUnit second = MeasureUnit::forIdentifier(first.getIdentifier(), status);
+
+    // Heap is freed here, as an old first.identifier CharString is deallocated
+    // and a new CharString is allocated:
+    first = first.product(crux, status);
+
+    // Proving we've had no failure yet:
+    if (U_FAILURE(status)) return;
+
+    // heap-use-after-free failure happened here, since a SingleUnitImpl had
+    // held onto a StringPiece pointing at a substring of an identifier that was
+    // freed above:
+    second = second.product(crux, status);
+}
+
 
 void MeasureFormatTest::verifyFieldPosition(
         const char *description,