The Hammers needed a penalty shoot-out to beat Maltese side Birkirkara on Thursday night having slipped to a 1-0 defeat on the evening.
Poyet came off the bench to score as all five of West Ham's penalty takers were successful following a gruelling 120 minutes in which James Tomkins was sent off for a tussle in the penalty area.
Birkirkara had won on the night courtesy of a goal from former Italy international Fabrizio Miccoli but did not do enough to find a second, decisive goal - settling instead for the lottery of the shoot-out, especially having been reduced to 10 men themselves in extra-time.
A third qualifying round tie with Romanian outfit Astra Giurgiu now awaits and Poyet wants a better performance from the team if they are to reach the play-off stage.
"The most important thing was to get through and we did it," he told the club's official website.
"It was obviously tough going a goal down and then going down to 10 men as well before half-time.
"We need to be careful, the further we progress the harder the games will be, so I don't know how many times we'll get away with it, but we're obviously very pleased to get through again."
Having seen off Birkirkara and Lusitans of Andorra in previous rounds, West Ham continue to come up against teams they know little about - and Poyet believes that could also cause problems as they face Astra - who saw off Inverness to set-up a meeting with the Hammers.
"Astra is another game in the unknown, we don't know what to expect," he added.
"We've got to learn from the two games, the things we've done well and the things we've done wrong, against this type of opposition.
"We want to win at home and away, get through and see what happens
We have to start off well, get ahead and take it to their place, but we need to be in the lead."
Source : PA
Source: PA