Chelsea coach Antonio Conte, who this month said his side were no longer in the title race given the dominance of runaway leaders Manchester City, said the Londoners were determined to fight for their place in next season's Champions League.
Morata, returning to the team after a one-match suspension, and Alonso both scored with headers to breach Brighton and Hove Albion's disciplined defence after a fallow first half.
Spearheaded by the lively Eden Hazard, Chelsea enjoyed the lion's share of possession and seldom looked vulnerable to a Brighton team who sought to thwart their hosts at every opportunity.
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Less than a minute into the second half, Morata sprang to meet Cesar Azpilicueta's diagonal ball, reviving an all-Spain partnership that has resulted in the defender providing six assists for the striker so far this season.
Chelsea wrapped up the match 13 minutes later when Alonso glanced in a near-post header from a corner by a fourth Spain international, Fabregas, to make it 2-0. It was Alonso's fifth league goal of the season.
The win increased pressure on second-placed Manchester United who could only manage a 2-2 home draw with Burnley.
Conte said there was "great motivation, to try to stay there, to try to fight until the end" but said the challenge to overhaul Manchester City's 13-point advantage over Chelsea - having played a game less than the Londoners - looked too great.
"We fight for a place in the Champions League next season. There are four places for six top teams," he said.
(Reporting by Clare Lovell and William Schomberg; Editing by Christian Radnedge and Pritha Sarkar)
