But after losing their opening home fixture, 40 overs of play were possible before the weather closed in.
It was dominated by Tom Westley and Cook following the loss of Nick Browne.
Their unbroken stand of 128 in 37 overs, with the stylish Westley making an unbeaten 70, fully justified Essex's decision to bat first, reaching 144-1.
He was the more aggressive batsmen, reaching his half century off 64 balls to brighten a gloomy, drizzly day.
It was the sixth successive first-class match in which he has passed 50, starting with his appearance for the MCC in Abu Dhabi against county champions Yorkshire in March.
England Test captain Cook, who has already scored two centuries this season against Gloucestershire and Sussex, hit seven fours and a six off England team-mate Moeen Ali as he completed a more sedate fifty off 112 balls.
Rain and bad light meant the loss of the entire afternoon session following a morning session in which Worcestershire's New Zealand international fast bowler Matt Henry went for 11 off his first over before quickly claiming the only wicket to fall when he had Browne caught behind.
Worcestershire assistant coach Matt Mason told BBC Hereford & Worcester:
"It was a frustrating day to watch as a coach because I know anything we do isn't through lack of effort.
"But we didn't quite have our discipline with us this morning and probably over-pitched and were a little bit on the full side with deliveries at times.
"Credit to them in some ways in trying to get those wickets early but they just didn't quite get it right and it cost us momentum really."
Essex head coach Chris Silverwood:
"We were delighted to win the toss. We got off to a good start. It is essential if you want to capitalise when you have won the toss.
"We have got ourselves in a reasonable position. I won't count any chickens yet. There is still along way to go.
"But hopefully we can restart tomorrow morning and build on the start the guys at the top of the order have given us and see where that takes us."

Isle of Wight Council tried to fine Jon Platt Â£120 for taking his daughter to Florida during the school term.
But magistrates backed the father, so the council appealed to the High Court, which again ruled in Mr Platt's favour.
The Department for Education is meeting the council's legal costs.
The legal case rests on what constitutes "regular attendance".
Mr Platt has argued that even after the "once-in-a-lifetime" family trip to Disneyland, his daughter had 90% attendance at school and this argument was accepted by the High Court in May last year.
At the time Lord Justice Lloyd Jones said the case "raised a point of law of general public importance".
After losing the case the council, at the request of Schools Minister Nick Gibb, applied to the Supreme Court for permission to launch a final legal challenge.
New guidelines came into force in England in 2013 making the policy on unauthorised term-time absences the toughest in the UK.
The change followed concerns that some families had started to see going away in term-time as an entitlement.
Under the rules, if a school declares an absence unauthorised, the council can fine each parent Â£60 per child - and this doubles to Â£120 if not paid within 21 days.
Councils in England have varying policies on term-time holiday fines.  Some issue thousands of fines - others barely any.
The case is forcing many to look again at the detail.
Most parents simply pay the fines to avoid prosecution, but  Mr Platt says thousands have contacted him for advice on fighting the fines.
Many parents complain that the cost of going away in the school holidays can be four times as much as during term-time - but the government says there is clear evidence "that every extra day of school missed can affect a pupil's chance of gaining good GCSEs, which has a lasting effect on their life chances".
A DfE spokeswoman said the government's position remained that children should not be taken out of school "without good reason".
"That is why we have tightened the rules and are supporting schools and local authorities to use their powers to tackle unauthorised absence."
Ministers are also considering changing the law on term-time absences, depending on the outcome of the case.
Judgement is expected during the first part of 2017.
England captain Alastair Cook continued his fine start to the season with an unbeaten 53 on another rain-hit day at New Road against Worcestershire.
[NEXT_CONCEPT]
A council which lost a High Court case over fining a father who took his daughter on a term-time holiday is having its appeal heard at the Supreme Court.