diff --git a/filebeat/input/syslog/input.go b/filebeat/input/syslog/input.go
index d8f7dde529a3..29b2325359fa 100644
--- a/filebeat/input/syslog/input.go
+++ b/filebeat/input/syslog/input.go
@@ -174,6 +174,7 @@ func (p *Input) Run() {
 		err := p.server.Start()
 		if err != nil {
 			p.log.Error("Error starting the server", "error", err)
+			return
 		}
 		p.started = true
 	}
@@ -185,6 +186,10 @@ func (p *Input) Stop() {
 	p.Lock()
 	defer p.Unlock()
 
+	if !p.started {
+		return
+	}
+
 	p.log.Info("Stopping Syslog input")
 	p.server.Stop()
 	p.started = false
diff --git a/filebeat/inputsource/udp/server.go b/filebeat/inputsource/udp/server.go
index 762e927a89bf..c263a2ca4e8a 100644
--- a/filebeat/inputsource/udp/server.go
+++ b/filebeat/inputsource/udp/server.go
@@ -96,7 +96,15 @@ func (u *Server) run() {
 				continue
 			}
 
-			u.log.Errorw("Error reading from the socket", "error", err)
+			// Closed network error string will never change in Go 1.X
+			// https://github.com/golang/go/issues/4373
+			opErr, ok := err.(*net.OpError)
+			if ok && strings.Contains(opErr.Err.Error(), "use of closed network connection") {
+				u.log.Info("Connection has been closed")
+				return
+			}
+
+			u.log.Errorf("Error reading from the socket %s", err)
 
 			// On Windows send the current buffer and mark it as truncated.
 			// The buffer will have content but length will return 0, addr will be nil.
@@ -115,8 +123,8 @@ func (u *Server) run() {
 // Stop stops the current udp server.
 func (u *Server) Stop() {
 	u.log.Info("Stopping UDP server")
-	u.Listener.Close()
 	close(u.done)
+	u.Listener.Close()
 	u.wg.Wait()
 	u.log.Info("UDP server stopped")
 }