Enner Valencia struck from inside the area after 53 seconds for the Hammers but Aguero equalised with a penalty after he had been fouled.
The hosts retook the lead when Valencia capitalised on hesitant defending by Nicolas Otamendi to stab in.
However, Aguero struck from 10 yards late on to earn his side a point.
The result means Manchester City are second, three points behind Leicester, while West Ham are sixth.
A good point, but City's away form a worry
Of the sides in the top three, Manchester City boast the worst away record.
They have won just four games on the road all season, while their last five away trips have seen them pick up just six points from a possible 18, a far from ideal record for a side challenging for the title.
Against West Ham, it was their inability to complete the basics of defending that was of particular concern.
They switched off in the first minute to allow West Ham to score, but their defending for the hosts' second was particularly poor.
Michail Antonio's quick throw initially caught out Otamendi, but he still had the time to be able to clear before Valencia nipped in to poke home.
Luckily for City they had Aguero to rescue them, but with away trips to struggling Sunderland and Newcastle coming up, Pellegrini's side can ill afford such lapses in those games.
Toure struggles to make an impression
There has been plenty of uncertainty about Yaya Toure's future in the newspapers this week, but City boss Manuel Pellegrini has urged the midfield to concentrate on helping his side challenge for the title.
Pellegrini clearly felt Toure is fully focused on the task at hand as he recalled him for the trip to West Ham, but was sluggish at the start.
He was left woefully wanting when Cheikhou Kouyate ghosted past inside a minute to set up Valencia for the opener.
In the end, Toure had more touches of the ball than any other City player, but struggled to make a telling impression in the opposition half, with only defenders Fernando and Otamendi having a worst passing accuracy than the midfielder.
Man of the match - Sergio Aguero
What next?
West Ham are in FA Cup fourth-round action at Liverpool on 30 January while Manchester City's next game is at Sunderland in the Premier League on 2 February.
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