diff --git a/src/mux_h1.c b/src/mux_h1.c
index 2bb532810..885269909 100644
--- a/src/mux_h1.c
+++ b/src/mux_h1.c
@@ -1568,7 +1568,10 @@ static size_t h1_process_input(struct h1c *h1c, struct buffer *buf, size_t count
 		h1s->cs->flags &= ~CS_FL_MAY_SPLICE;
 	}
 
-	if (h1s->flags & H1S_F_PARSING_DONE)
+	/* Don't set EOI on the conn-stream for protocol upgrade requests, wait
+	 * the response to do so or not depending on the status code.
+	 */
+	if ((h1s->flags & H1S_F_PARSING_DONE) && !(h1m->flags & H1_MF_CONN_UPG))
 		h1s->cs->flags |= CS_FL_EOI;
 
 	if (h1s_data_pending(h1s) && !htx_is_empty(htx))
@@ -1577,8 +1580,10 @@ static size_t h1_process_input(struct h1c *h1c, struct buffer *buf, size_t count
 		h1s->cs->flags &= ~(CS_FL_RCV_MORE | CS_FL_WANT_ROOM);
 		if (h1s->flags & H1S_F_REOS) {
 			h1s->cs->flags |= CS_FL_EOS;
-			if (h1m->state == H1_MSG_TUNNEL)
+			if (h1m->state >= H1_MSG_DONE) {
+				/* DONE or TUNNEL, set EOI on the conn-stream */
 				h1s->cs->flags |= CS_FL_EOI;
+			}
 			else if (h1m->state > H1_MSG_LAST_LF && h1m->state < H1_MSG_DONE)
 				h1s->cs->flags |= CS_FL_ERROR;
 		}
@@ -2023,6 +2028,13 @@ static size_t h1_process_output(struct h1c *h1c, struct buffer *buf, size_t coun
 
 	htx_to_buf(chn_htx, buf);
   out:
+	/* Both the request and the response reached the DONE state. So set EOI
+	 * flag on the conn-stream. Most of time, the flag will already be set,
+	 * except for protocol upgrades.
+	 */
+	if (h1s->cs && h1s->req.state == H1_MSG_DONE && h1s->res.state == H1_MSG_DONE)
+			h1s->cs->flags |= CS_FL_EOI;
+
 	if (!buf_room_for_htx_data(&h1c->obuf)) {
 		TRACE_STATE("h1c obuf full", H1_EV_TX_DATA|H1_EV_H1S_BLK, h1c->conn, h1s);
 		h1c->flags |= H1C_F_OUT_FULL;