There will be a ban on the possession, transport, selling or growing of species deemed as of "Union Concern".
The list was going to be restricted to 50, but will now have no limit. It is not clear which species will be banned.
A deal between EU member states effectively means the bill will pass and become law within a few months.
This comes as MPs slam current government policy on controlling alien species as "not fit for purpose".
Biodiversity lossThe economic and ecological damage caused by non-native species such as the so-called killer shrimp and demon shrimp originally from the Black Sea, the Asian Harlequin ladybird or Japanese knotweed are estimated to cost Europe some 12bn euros every year.
Continue reading the main storyEfforts are very often not effective simply because those species do not respect geographical boundaries”End Quote Pavel Poc MEP In the UK the bill is at least £1.7bn.
Experts say such insects, plants and animals are one of the major causes of biodiversity loss and species extinction.
They can also spread disease and cause health problems such as asthma, dermatitis and allergies.
The new law will require member states to analyse how troublesome species enter the country and to improve surveillance systems.
Official checks at EU borders will be stepped up. Action plans on how to manage established invasive species also have to be developed.
MEP Pavel Poc, who is guiding the legislation through the European Parliament, said: "Efforts are very often not effective simply because those species do not respect geographical boundaries. Co-operation between member states is therefore crucial."
Garden plantsWednesday's report from the Environmental Audit Committee welcomes the strengthening of the rules.
Committee chairwoman Joan Walley MP told the BBC: "The UK has to be ready to take on board the step changes that there will be as a result of the European decision.
"People are travelling more, and international trade means there are all kinds of opportunities which there previously weren't for non-native species to come in to the UK.
Brazilian rhubarb is an invasive species that likes boggy land 
Certain types of rhododendron are highly invasive "We need to have a regime that is fit for purpose to deal with this before it gets out of hand."
While welcoming the overall aim of the legislation, the Royal Horticultural Society (RHS) is concerned about the lack of clarity around the new powers.
Guy Barter, RHS chief horticultural adviser, said: "There is a rhododendron that is highly invasive. It is called Rhododendron ponticum. It would be a candidate for any list of prohibited plants, but we reckon it is in the parentage of as many as 300 garden plants.
"So if the legislation included Rhododendron ponticum and its hybrids, that would affect 300 useful garden plants. This is certainly a concern. It would mean that gardens would be less rich as gardeners wouldn't be able to buy these plants."
Voracious predatorsThe Department for Environment Food and Rural Affairs (Defra) currently has its own guidelines on restricted species.
Defra has made assurances that member states will have flexibility in how they implement these rules.
A Defra spokesperson said: "This report recognises we have the most advanced approach across Europe in tackling the threat of invasive non-native species, helping identify and prevent the spread of plants and animals that don't belong here."
There are an estimated 12,000 alien species in Europe and 2,000 in Britain.
Continue reading the main storyWe need to be doing a lot more to tackle the control of species already causing havoc within our UK streams, rivers and estuaries and harbours”End Quote Dr Alex Ford Institute of Marine Sciences, University of Portsmouth However only a minority are invasive and have seriously negative impacts. For example wheat or barley or sheep are not native to the UK and do not cause problems.
Relative newcomers like killer and demon shrimp (Dikerogammarus villosus and Dikerogammarus haemobaphes) are voracious predators.
It has been fewer than five years since they were found in England but they have already spread to lakes and waterways in many different parts of the country.
Lucy Anderson from the University of Leeds led a research project into killer shrimp.
She found that they hitchhike in the kit of fishermen and canoeists and spread to new waterways. They can survive out of the water for up to two weeks.
"Invasive species can have a huge impact on our native species. They can out-compete them for food and for resources, they can also introduce diseases which our native species aren't resistant to, so wiping them out completely. It's a huge problem," she said.
'Considerable damage'Her colleague Dr Alison Dunn added: "One of the important things we need to do is stop them spreading further.
"We would encourage water users when they have been somewhere to stop when they have finished and check their equipment to see there are not any of these invaders on it."
Dr Alison Dunn and Lucy Anderson have conducted a research project into killer shrimp A new investigation currently being carried out by the university seems to show that killer shrimp are killed off by water from a hot tap. This would provide a useful tool in the fight to contain the species.
Demon shrimp have been studied extensively by Dr Alex Ford, from the Institute of Marine Sciences at the University of Portsmouth.
"Certainly from the aquatic environment we need to be doing a lot more to tackle the control of species already causing havoc within our UK streams, rivers and estuaries and harbours.
"In addition, we need to be doing a lot more to prevent them being introduced in the first place and giving them the ability to take hold.
"Species such as American crayfish, demon and killer shrimp are causing considerable damage to British ecosystems and we don't currently know whether they also pose a biosecurity threat in terms of parasites and disease they being with them."
