Captain Cooper started against the Saints and was solid defensively as his side dominated most of the first half and took a deserved lead through Jack Harrison.
Southampton worked their way back into the game in the latter stages of the first half and then came out flying following the interval, a sublime James Ward-Prowse free-kick levelling matters.
Both sides had chances to win it, Leeds recovering from their poor start to the second half as Phillips came off the bench to calm the home side down but there was to be no victor and the spoils were shared.
Leeds are now unbeaten in three and sit eight points clear of the drop zone thanks to Watford's 2-0 defeat by Liverpool earlier in the day.
7. Mateusz Klich - 6
Some clever bits of play that helped Leeds attack in the first half, made a nuisance of himself but faded just a little at the start of the second.
8. Jack Harrison - 5
Willing to run all day, always showed himself as an option but apart from the goal struggled to do the right things at the right time.
9. Raphinha - 7
Carried a threat throughout, did brilliantly to set up the goal, unlucky once or twice in the final third but still not at his absolute best.
10. Rodrigo - 8
Best player on the pitch at times. Lots of really clever play to start attacks or find team-mates. Tired towards the very end but kept going.
11. Daniel James - 6
Gave Leeds a threat with pace but lacked the composure and decision-making that would have produced chances and maybe goals.
12. Joe Gelhardt - 7
Bright, lively, determined. Got straight into it and gave Leeds a bit more presence up top. Took up good positions without being found.