diff --git a/h2quic/roundtrip.go b/h2quic/roundtrip.go
index e725ea7a..6b195b6f 100644
--- a/h2quic/roundtrip.go
+++ b/h2quic/roundtrip.go
@@ -11,6 +11,8 @@ import (
 
 	quic "github.com/lucas-clemente/quic-go"
 
+	"runtime"
+
 	"golang.org/x/net/lex/httplex"
 )
 
@@ -91,6 +93,7 @@ func (r *RoundTripper) getClient(hostname string) http.RoundTripper {
 	defer r.mutex.Unlock()
 
 	if r.clients == nil {
+		runtime.SetFinalizer(r, finalizer)
 		r.clients = make(map[string]roundTripCloser)
 	}
 
@@ -102,6 +105,10 @@ func (r *RoundTripper) getClient(hostname string) http.RoundTripper {
 	return client
 }
 
+func finalizer(r *RoundTripper) {
+	_ = r.Close()
+}
+
 // Close closes the QUIC connections that this RoundTripper has used
 func (r *RoundTripper) Close() error {
 	r.mutex.Lock()
diff --git a/h2quic/roundtrip_test.go b/h2quic/roundtrip_test.go
index b612d8bb..66e3eb8a 100644
--- a/h2quic/roundtrip_test.go
+++ b/h2quic/roundtrip_test.go
@@ -6,6 +6,7 @@ import (
 	"errors"
 	"io"
 	"net/http"
+	"runtime"
 	"time"
 
 	quic "github.com/lucas-clemente/quic-go"
@@ -196,5 +197,16 @@ var _ = Describe("RoundTripper", func() {
 			Expect(err).ToNot(HaveOccurred())
 			Expect(len(rt.clients)).To(BeZero())
 		})
+
+		It("runs Close when the RoundTripper is garbage collected", func() {
+			// this is set by getClient, but we can't do that while at the same time injecting the mockClient
+			runtime.SetFinalizer(rt, finalizer)
+			rt.clients = make(map[string]roundTripCloser)
+			cl := &mockClient{}
+			rt.clients["foo.bar"] = cl
+			rt = nil // lose the references to the RoundTripper, such that it can be garbage collected
+			runtime.GC()
+			Eventually(func() bool { return cl.closed }).Should(BeTrue())
+		})
 	})
 })