Vault Agent Injector cant communicate with Kubernetes inside Istio Mesh

I’ve deployed Vault Sidecar following Injecting Secrets into Kubernetes Pods via Vault Helm Sidecar | Vault - HashiCorp Learn but the vault-agent-injector can’t communicate with Kubernetes, giving the following error:
Error updating MutatingWebhookConfiguration: Patch https://10.245.0.1:443/apis/admissionregistration.k8s.io/v1beta1/mutatingwebhookconfigurations/vault-agent-injector-cfg: dial tcp 10.245.0.1:443: connect: connection refused

I’ve figured that error was probably due to Istio Proxy sidecar being present on the same pod, so I configured a ServiceEntry like that:

apiVersion: networking.istio.io/v1alpha3
kind: ServiceEntry
metadata:
  name: vault-service-entry
  namespace: vault
spec:
  hosts:
    - "kubernetes.default.svc.cluster.local"
  ports:
    - number: 443
      name: https
      protocol: HTTPS
  location: MESH_EXTERNAL
  resolution: DNS

But now I’m getting:

Error updating MutatingWebhookConfiguration: Patch https://10.245.0.1:443/apis/admissionregistration.k8s.io/v1beta1/mutatingwebhookconfigurations/vault-agent-injector-cfg: read tcp 10.244.1.196:56774->10.245.0.1:443: read: connection reset by peer
Error updating MutatingWebhookConfiguration: Patch https://10.245.0.1:443/apis/admissionregistration.k8s.io/v1beta1/mutatingwebhookconfigurations/vault-agent-injector-cfg: EOF

Was anybody able to make them (istio + vault) work well together?