diff --git a/core/src/main/scala/org/apache/spark/scheduler/TaskSetManager.scala b/core/src/main/scala/org/apache/spark/scheduler/TaskSetManager.scala
index a18c66596852a..415af893cbc49 100644
--- a/core/src/main/scala/org/apache/spark/scheduler/TaskSetManager.scala
+++ b/core/src/main/scala/org/apache/spark/scheduler/TaskSetManager.scala
@@ -723,6 +723,12 @@ private[spark] class TaskSetManager(
   def handleSuccessfulTask(tid: Long, result: DirectTaskResult[_]): Unit = {
     val info = taskInfos(tid)
     val index = info.index
+    // Check if any other attempt succeeded before this and this attempt has not been handled
+    if (successful(index) && killedByOtherAttempt(index)) {
+      handleFailedTask(tid, TaskState.KILLED, TaskKilled("another attempt succeeded"))
+      return
+    }
+
     info.markFinished(TaskState.FINISHED, clock.getTimeMillis())
     if (speculationEnabled) {
       successfulTaskDurations.insert(info.duration)
diff --git a/core/src/test/scala/org/apache/spark/scheduler/TaskSetManagerSuite.scala b/core/src/test/scala/org/apache/spark/scheduler/TaskSetManagerSuite.scala
index ca6a7e5db3b17..c0b418bd219a9 100644
--- a/core/src/test/scala/org/apache/spark/scheduler/TaskSetManagerSuite.scala
+++ b/core/src/test/scala/org/apache/spark/scheduler/TaskSetManagerSuite.scala
@@ -1371,4 +1371,64 @@ class TaskSetManagerSuite extends SparkFunSuite with LocalSparkContext with Logg
     val valueSer = SparkEnv.get.serializer.newInstance()
     new DirectTaskResult[Int](valueSer.serialize(id), accumUpdates)
   }
+
+  test("SPARK-13343 speculative tasks that didn't commit shouldn't be marked as success"){
+    sc = new SparkContext("local", "test")
+    sched = new FakeTaskScheduler(sc, ("exec1", "host1"), ("exec2", "host2"))
+    val taskSet = FakeTask.createTaskSet(4)
+    // Set the speculation multiplier to be 0 so speculative tasks are launched immediately
+    sc.conf.set("spark.speculation.multiplier", "0.0")
+    sc.conf.set("spark.speculation", "true")
+    val clock = new ManualClock()
+    val manager = new TaskSetManager(sched, taskSet, MAX_TASK_FAILURES, clock = clock)
+    val accumUpdatesByTask: Array[Seq[AccumulatorV2[_, _]]] = taskSet.tasks.map { task =>
+      task.metrics.internalAccums
+    }
+
+    // Offer resources for 4 tasks to start
+    for ((k, v) <- List(
+      "exec1" -> "host1",
+      "exec1" -> "host1",
+      "exec2" -> "host2",
+      "exec2" -> "host2")) {
+      val taskOption = manager.resourceOffer(k, v, NO_PREF)
+      assert(taskOption.isDefined)
+      val task = taskOption.get
+      assert(task.executorId === k)
+    }
+    assert(sched.startedTasks.toSet === Set(0, 1, 2, 3))
+    clock.advance(1)
+    // Complete the 3 tasks and leave 1 task in running
+    for (id <- Set(0, 1, 2)) {
+      manager.handleSuccessfulTask(id, createTaskResult(id, accumUpdatesByTask(id)))
+      assert(sched.endedTasks(id) === Success)
+    }
+
+    // checkSpeculatableTasks checks that the task runtime is greater than the threshold for
+    // speculating. Since we use a threshold of 0 for speculation, tasks need to be running for
+    // > 0ms, so advance the clock by 1ms here.
+    clock.advance(1)
+    assert(manager.checkSpeculatableTasks(0))
+    assert(sched.speculativeTasks.toSet === Set(3))
+
+    // Offer resource to start the speculative attempt for the running task
+    val taskOption5 = manager.resourceOffer("exec1", "host1", NO_PREF)
+    assert(taskOption5.isDefined)
+    val task5 = taskOption5.get
+    assert(task5.index === 3)
+    assert(task5.taskId === 4)
+    assert(task5.executorId === "exec1")
+    assert(task5.attemptNumber === 1)
+    sched.backend = mock(classOf[SchedulerBackend])
+
+    // Complete one attempt for the running task
+    manager.handleSuccessfulTask(3, createTaskResult(3, accumUpdatesByTask(3)))
+    // Verify that it kills other running attempt
+    verify(sched.backend).killTask(4, "exec1", true, "another attempt succeeded")
+    // Complete another attempt for the running task
+    manager.handleSuccessfulTask(4, createTaskResult(3, accumUpdatesByTask(3)))
+
+    assert(manager.taskInfos(3).successful == true)
+    assert(manager.taskInfos(4).killed == true)
+  }
 }