United stretched their winless run to eight games despite a much improved attacking performance in the final match of 2015.
Both sides had clear opportunities to nick the points, but a draw was ultimately a fair result from an engrossing encounter.
The hosts were on the attack from the off and almost went ahead after only three minutes when Juan Mata came within inches against his former club when he rattled the crossbar with a stinging drive.
John Terry forced an excellent save from David de Gea from Eden Hazard's corner, whilst the recalled Morgan Schneiderlin flashed an effort narrowly wide at the other end.
Anthony Martial fired against the post after superb trickery against Branislav Ivanovic, Rooney was kept out by a diving save from Courtois and Ashley Young brilliantly blocked from Willian as Chelsea broke away at pace.
The sides remained deadlocked at the break, but it was the visitors who were quicker out of the traps and went close twice soon after the restart.
Hazard cleverly weaved his way through the United defence and teed up Pedro, who shot was parried clear by De Gea but the rebound fell to Cesar Azpilicueta only for De Gea to paw away the full back's first time effort.
A huge penalty shout soon followed for United as Willian appeared to handle from a corner, but referee Martin Atkinson remained unmoved.
LVG's side were soon appealing for another spot-kick after Mata went down under a challenge from Azpilicueta, but not before Nemanja Matic had spurned probably the best chance of the match- blazing over when clean through after a rapid Chelsea break.
Courtois saved brilliantly from the busy Ander Herrera from point blank range after more excellent work from Martial shortly before the hour mark.
The Reds continued to press and fashioned another presentable opening in the dying minutes when Wayne Rooney volleyed over a difficult chance from substitute Cameron Borthwick-Jackson's pinpoint cross.
Rooney was perhaps then fortunate to escape punishment after a scuffle with Oscar, but despite late forward forays from both sides, ultimately the breakthrough remained elusive and the spoils were shared.
A frustrating result - it's now no win in eight - but a much improved performance and one that will restore confidence to the side ahead of the home match with recent bogey side Swansea on Saturday.
Overall team performance: 7.5/10
United Faithful Man of the Match: A number of outstanding performers in Red, but we're giving it to David de Gea for three excellent and crucial saves.